One LP Convention, Exact Slack Maps and Mode-Specific Conversions
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[1, 1], [-1, 2]], b=[4, 2], c=[3, -1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 1], [-1, 2]], [4, 2], [3, -1], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[1, 1], [-1, 2]], [4, 2], [3, -1], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
2. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[2, 1], [-1, 3]], b=[6, 3], c=[4, -2]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[2, 1], [-1, 3]], [6, 3], [4, -2], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[2, 1], [-1, 3]], [6, 3], [4, -2], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
3. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[3, 1], [-1, 4]], b=[8, 4], c=[5, -3]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[3, 1], [-1, 4]], [8, 4], [5, -3], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[3, 1], [-1, 4]], [8, 4], [5, -3], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
4. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[4, 1], [-1, 5]], b=[10, 5], c=[6, -4]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[4, 1], [-1, 5]], [10, 5], [6, -4], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[4, 1], [-1, 5]], [10, 5], [6, -4], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
5. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[5, 1], [-1, 6]], b=[12, 6], c=[7, -5]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[5, 1], [-1, 6]], [12, 6], [7, -5], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[5, 1], [-1, 6]], [12, 6], [7, -5], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
6. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[6, 1], [-1, 7]], b=[14, 7], c=[8, -6]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[6, 1], [-1, 7]], [14, 7], [8, -6], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[6, 1], [-1, 7]], [14, 7], [8, -6], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
7. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[7, 1], [-1, 8]], b=[16, 8], c=[9, -7]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[7, 1], [-1, 8]], [16, 8], [9, -7], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[7, 1], [-1, 8]], [16, 8], [9, -7], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
8. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[8, 1], [-1, 9]], b=[18, 9], c=[10, -8]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[8, 1], [-1, 9]], [18, 9], [10, -8], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[8, 1], [-1, 9]], [18, 9], [10, -8], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
9. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[9, 1], [-1, 10]], b=[20, 10], c=[11, -9]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[9, 1], [-1, 10]], [20, 10], [11, -9], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[9, 1], [-1, 10]], [20, 10], [11, -9], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
10. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[10, 1], [-1, 11]], b=[22, 11], c=[12, -10]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[10, 1], [-1, 11]], [22, 11], [12, -10], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[10, 1], [-1, 11]], [22, 11], [12, -10], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
11. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[11, 1], [-1, 12]], b=[24, 12], c=[13, -11]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[11, 1], [-1, 12]], [24, 12], [13, -11], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[11, 1], [-1, 12]], [24, 12], [13, -11], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
12. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[12, 1], [-1, 13]], b=[26, 13], c=[14, -12]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[12, 1], [-1, 13]], [26, 13], [14, -12], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[12, 1], [-1, 13]], [26, 13], [14, -12], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
13. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[13, 1], [-1, 14]], b=[28, 14], c=[15, -13]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[13, 1], [-1, 14]], [28, 14], [15, -13], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[13, 1], [-1, 14]], [28, 14], [15, -13], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
14. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[14, 1], [-1, 15]], b=[30, 15], c=[16, -14]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[14, 1], [-1, 15]], [30, 15], [16, -14], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[14, 1], [-1, 15]], [30, 15], [16, -14], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
15. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[15, 1], [-1, 16]], b=[32, 16], c=[17, -15]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[15, 1], [-1, 16]], [32, 16], [17, -15], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[15, 1], [-1, 16]], [32, 16], [17, -15], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
16. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[16, 1], [-1, 17]], b=[34, 17], c=[18, -16]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[16, 1], [-1, 17]], [34, 17], [18, -16], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[16, 1], [-1, 17]], [34, 17], [18, -16], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
17. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[17, 1], [-1, 18]], b=[36, 18], c=[19, -17]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[17, 1], [-1, 18]], [36, 18], [19, -17], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[17, 1], [-1, 18]], [36, 18], [19, -17], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
18. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[18, 1], [-1, 19]], b=[38, 19], c=[20, -18]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[18, 1], [-1, 19]], [38, 19], [20, -18], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[18, 1], [-1, 19]], [38, 19], [20, -18], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
19. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[19, 1], [-1, 20]], b=[40, 20], c=[21, -19]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[19, 1], [-1, 20]], [40, 20], [21, -19], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[19, 1], [-1, 20]], [40, 20], [21, -19], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
20. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[20, 1], [-1, 21]], b=[42, 21], c=[22, -20]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[20, 1], [-1, 21]], [42, 21], [22, -20], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[20, 1], [-1, 21]], [42, 21], [22, -20], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
21. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[21, 1], [-1, 22]], b=[44, 22], c=[23, -21]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[21, 1], [-1, 22]], [44, 22], [23, -21], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[21, 1], [-1, 22]], [44, 22], [23, -21], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
22. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[22, 1], [-1, 23]], b=[46, 23], c=[24, -22]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[22, 1], [-1, 23]], [46, 23], [24, -22], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[22, 1], [-1, 23]], [46, 23], [24, -22], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
23. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[23, 1], [-1, 24]], b=[48, 24], c=[25, -23]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[23, 1], [-1, 24]], [48, 24], [25, -23], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[23, 1], [-1, 24]], [48, 24], [25, -23], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
24. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[24, 1], [-1, 25]], b=[50, 25], c=[26, -24]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[24, 1], [-1, 25]], [50, 25], [26, -24], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[24, 1], [-1, 25]], [50, 25], [26, -24], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
25. Qualify Dimensions and the Fixed Convention without Claiming Feasibility
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[25, 1], [-1, 26]], b=[52, 26], c=[27, -25]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[25, 1], [-1, 26]], [52, 26], [27, -25], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)
Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[25, 1], [-1, 26]], [52, 26], [27, -25], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-dimension-sign-qualification · candidate
26. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[1, 1], [1, 1]], b=[4, 4], c=[2, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [2, 2], [4, 4], [2, 2], Yes, Yes, Yes, Yes, Yes, 3, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [2, 2], [4, 4], [2, 2], Yes, Yes, Yes, Yes, Yes, 3, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
27. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[2, 1], [1, 2]], b=[5, 5], c=[3, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [3, 3], [5, 5], [2, 2], Yes, Yes, Yes, Yes, Yes, 4, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [3, 3], [5, 5], [2, 2], Yes, Yes, Yes, Yes, Yes, 4, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
28. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[3, 1], [1, 3]], b=[6, 6], c=[4, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [4, 4], [6, 6], [2, 2], Yes, Yes, Yes, Yes, Yes, 5, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [4, 4], [6, 6], [2, 2], Yes, Yes, Yes, Yes, Yes, 5, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
29. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[4, 1], [1, 4]], b=[7, 7], c=[5, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [5, 5], [7, 7], [2, 2], Yes, Yes, Yes, Yes, Yes, 6, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [5, 5], [7, 7], [2, 2], Yes, Yes, Yes, Yes, Yes, 6, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
30. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[5, 1], [1, 5]], b=[8, 8], c=[6, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [6, 6], [8, 8], [2, 2], Yes, Yes, Yes, Yes, Yes, 7, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [6, 6], [8, 8], [2, 2], Yes, Yes, Yes, Yes, Yes, 7, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
31. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[6, 1], [1, 6]], b=[9, 9], c=[7, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [7, 7], [9, 9], [2, 2], Yes, Yes, Yes, Yes, Yes, 8, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [7, 7], [9, 9], [2, 2], Yes, Yes, Yes, Yes, Yes, 8, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
32. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[7, 1], [1, 7]], b=[10, 10], c=[8, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [8, 8], [10, 10], [2, 2], Yes, Yes, Yes, Yes, Yes, 9, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [8, 8], [10, 10], [2, 2], Yes, Yes, Yes, Yes, Yes, 9, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
33. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[8, 1], [1, 8]], b=[11, 11], c=[9, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [9, 9], [11, 11], [2, 2], Yes, Yes, Yes, Yes, Yes, 10, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [9, 9], [11, 11], [2, 2], Yes, Yes, Yes, Yes, Yes, 10, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
34. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[9, 1], [1, 9]], b=[12, 12], c=[10, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [10, 10], [12, 12], [2, 2], Yes, Yes, Yes, Yes, Yes, 11, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [10, 10], [12, 12], [2, 2], Yes, Yes, Yes, Yes, Yes, 11, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
35. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[10, 1], [1, 10]], b=[13, 13], c=[11, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [11, 11], [13, 13], [2, 2], Yes, Yes, Yes, Yes, Yes, 12, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [11, 11], [13, 13], [2, 2], Yes, Yes, Yes, Yes, Yes, 12, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
36. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[11, 1], [1, 11]], b=[14, 14], c=[12, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [12, 12], [14, 14], [2, 2], Yes, Yes, Yes, Yes, Yes, 13, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [12, 12], [14, 14], [2, 2], Yes, Yes, Yes, Yes, Yes, 13, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
37. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[12, 1], [1, 12]], b=[15, 15], c=[13, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [13, 13], [15, 15], [2, 2], Yes, Yes, Yes, Yes, Yes, 14, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [13, 13], [15, 15], [2, 2], Yes, Yes, Yes, Yes, Yes, 14, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
38. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[13, 1], [1, 13]], b=[16, 16], c=[14, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [14, 14], [16, 16], [2, 2], Yes, Yes, Yes, Yes, Yes, 15, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [14, 14], [16, 16], [2, 2], Yes, Yes, Yes, Yes, Yes, 15, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
39. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[14, 1], [1, 14]], b=[17, 17], c=[15, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [15, 15], [17, 17], [2, 2], Yes, Yes, Yes, Yes, Yes, 16, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [15, 15], [17, 17], [2, 2], Yes, Yes, Yes, Yes, Yes, 16, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
40. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[15, 1], [1, 15]], b=[18, 18], c=[16, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [16, 16], [18, 18], [2, 2], Yes, Yes, Yes, Yes, Yes, 17, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [16, 16], [18, 18], [2, 2], Yes, Yes, Yes, Yes, Yes, 17, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
41. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[16, 1], [1, 16]], b=[19, 19], c=[17, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [17, 17], [19, 19], [2, 2], Yes, Yes, Yes, Yes, Yes, 18, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [17, 17], [19, 19], [2, 2], Yes, Yes, Yes, Yes, Yes, 18, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
42. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[17, 1], [1, 17]], b=[20, 20], c=[18, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [18, 18], [20, 20], [2, 2], Yes, Yes, Yes, Yes, Yes, 19, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [18, 18], [20, 20], [2, 2], Yes, Yes, Yes, Yes, Yes, 19, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
43. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[18, 1], [1, 18]], b=[21, 21], c=[19, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [19, 19], [21, 21], [2, 2], Yes, Yes, Yes, Yes, Yes, 20, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [19, 19], [21, 21], [2, 2], Yes, Yes, Yes, Yes, Yes, 20, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
44. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[19, 1], [1, 19]], b=[22, 22], c=[20, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [20, 20], [22, 22], [2, 2], Yes, Yes, Yes, Yes, Yes, 21, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [20, 20], [22, 22], [2, 2], Yes, Yes, Yes, Yes, Yes, 21, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
45. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[20, 1], [1, 20]], b=[23, 23], c=[21, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [21, 21], [23, 23], [2, 2], Yes, Yes, Yes, Yes, Yes, 22, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [21, 21], [23, 23], [2, 2], Yes, Yes, Yes, Yes, Yes, 22, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
46. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[21, 1], [1, 21]], b=[24, 24], c=[22, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [22, 22], [24, 24], [2, 2], Yes, Yes, Yes, Yes, Yes, 23, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [22, 22], [24, 24], [2, 2], Yes, Yes, Yes, Yes, Yes, 23, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
47. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[22, 1], [1, 22]], b=[25, 25], c=[23, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [23, 23], [25, 25], [2, 2], Yes, Yes, Yes, Yes, Yes, 24, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [23, 23], [25, 25], [2, 2], Yes, Yes, Yes, Yes, Yes, 24, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
48. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[23, 1], [1, 23]], b=[26, 26], c=[24, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [24, 24], [26, 26], [2, 2], Yes, Yes, Yes, Yes, Yes, 25, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [24, 24], [26, 26], [2, 2], Yes, Yes, Yes, Yes, Yes, 25, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
49. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[24, 1], [1, 24]], b=[27, 27], c=[25, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [25, 25], [27, 27], [2, 2], Yes, Yes, Yes, Yes, Yes, 26, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [25, 25], [27, 27], [2, 2], Yes, Yes, Yes, Yes, Yes, 26, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
50. The Slack Equality Representation Preserves Feasibility in Both Directions
Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[25, 1], [1, 25]], b=[28, 28], c=[26, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).
View answer and explanation
([1, 1], [26, 26], [28, 28], [2, 2], Yes, Yes, Yes, Yes, Yes, 27, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)
Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [26, 26], [28, 28], [2, 2], Yes, Yes, Yes, Yes, Yes, 27, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-slack-conversion · candidate
51. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[1, -2, 3], objective coefficient=4, unrestricted value=1. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([1, -2, 3], 4, 1, 1, 0, 1, [1, -2, 3], [1, -2, 3], Yes, 4, 4, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([1, -2, 3], 4, 1, 1, 0, 1, [1, -2, 3], [1, -2, 3], Yes, 4, 4, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
52. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[2, -3, 4], objective coefficient=5, unrestricted value=-2. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([2, -3, 4], 5, -2, 0, 2, -2, [-4, 6, -8], [-4, 6, -8], Yes, -10, -10, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([2, -3, 4], 5, -2, 0, 2, -2, [-4, 6, -8], [-4, 6, -8], Yes, -10, -10, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
53. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[3, -4, 5], objective coefficient=6, unrestricted value=3. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([3, -4, 5], 6, 3, 3, 0, 3, [9, -12, 15], [9, -12, 15], Yes, 18, 18, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([3, -4, 5], 6, 3, 3, 0, 3, [9, -12, 15], [9, -12, 15], Yes, 18, 18, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
54. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[4, -5, 6], objective coefficient=7, unrestricted value=-4. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([4, -5, 6], 7, -4, 0, 4, -4, [-16, 20, -24], [-16, 20, -24], Yes, -28, -28, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([4, -5, 6], 7, -4, 0, 4, -4, [-16, 20, -24], [-16, 20, -24], Yes, -28, -28, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
55. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[5, -6, 7], objective coefficient=8, unrestricted value=5. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([5, -6, 7], 8, 5, 5, 0, 5, [25, -30, 35], [25, -30, 35], Yes, 40, 40, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([5, -6, 7], 8, 5, 5, 0, 5, [25, -30, 35], [25, -30, 35], Yes, 40, 40, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
56. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[6, -7, 8], objective coefficient=9, unrestricted value=-6. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([6, -7, 8], 9, -6, 0, 6, -6, [-36, 42, -48], [-36, 42, -48], Yes, -54, -54, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([6, -7, 8], 9, -6, 0, 6, -6, [-36, 42, -48], [-36, 42, -48], Yes, -54, -54, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
57. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[7, -8, 9], objective coefficient=10, unrestricted value=7. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([7, -8, 9], 10, 7, 7, 0, 7, [49, -56, 63], [49, -56, 63], Yes, 70, 70, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([7, -8, 9], 10, 7, 7, 0, 7, [49, -56, 63], [49, -56, 63], Yes, 70, 70, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
58. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[8, -9, 10], objective coefficient=11, unrestricted value=-8. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([8, -9, 10], 11, -8, 0, 8, -8, [-64, 72, -80], [-64, 72, -80], Yes, -88, -88, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([8, -9, 10], 11, -8, 0, 8, -8, [-64, 72, -80], [-64, 72, -80], Yes, -88, -88, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
59. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[9, -10, 11], objective coefficient=12, unrestricted value=9. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([9, -10, 11], 12, 9, 9, 0, 9, [81, -90, 99], [81, -90, 99], Yes, 108, 108, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([9, -10, 11], 12, 9, 9, 0, 9, [81, -90, 99], [81, -90, 99], Yes, 108, 108, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
60. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[10, -11, 12], objective coefficient=13, unrestricted value=-10. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([10, -11, 12], 13, -10, 0, 10, -10, [-100, 110, -120], [-100, 110, -120], Yes, -130, -130, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([10, -11, 12], 13, -10, 0, 10, -10, [-100, 110, -120], [-100, 110, -120], Yes, -130, -130, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
61. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[11, -12, 13], objective coefficient=14, unrestricted value=11. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([11, -12, 13], 14, 11, 11, 0, 11, [121, -132, 143], [121, -132, 143], Yes, 154, 154, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([11, -12, 13], 14, 11, 11, 0, 11, [121, -132, 143], [121, -132, 143], Yes, 154, 154, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
62. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[12, -13, 14], objective coefficient=15, unrestricted value=-12. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([12, -13, 14], 15, -12, 0, 12, -12, [-144, 156, -168], [-144, 156, -168], Yes, -180, -180, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([12, -13, 14], 15, -12, 0, 12, -12, [-144, 156, -168], [-144, 156, -168], Yes, -180, -180, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
63. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[13, -14, 15], objective coefficient=16, unrestricted value=13. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([13, -14, 15], 16, 13, 13, 0, 13, [169, -182, 195], [169, -182, 195], Yes, 208, 208, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([13, -14, 15], 16, 13, 13, 0, 13, [169, -182, 195], [169, -182, 195], Yes, 208, 208, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
64. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[14, -15, 16], objective coefficient=17, unrestricted value=-14. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([14, -15, 16], 17, -14, 0, 14, -14, [-196, 210, -224], [-196, 210, -224], Yes, -238, -238, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([14, -15, 16], 17, -14, 0, 14, -14, [-196, 210, -224], [-196, 210, -224], Yes, -238, -238, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
65. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[15, -16, 17], objective coefficient=18, unrestricted value=15. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([15, -16, 17], 18, 15, 15, 0, 15, [225, -240, 255], [225, -240, 255], Yes, 270, 270, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([15, -16, 17], 18, 15, 15, 0, 15, [225, -240, 255], [225, -240, 255], Yes, 270, 270, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
66. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[16, -17, 18], objective coefficient=19, unrestricted value=-16. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([16, -17, 18], 19, -16, 0, 16, -16, [-256, 272, -288], [-256, 272, -288], Yes, -304, -304, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([16, -17, 18], 19, -16, 0, 16, -16, [-256, 272, -288], [-256, 272, -288], Yes, -304, -304, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
67. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[17, -18, 19], objective coefficient=20, unrestricted value=17. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([17, -18, 19], 20, 17, 17, 0, 17, [289, -306, 323], [289, -306, 323], Yes, 340, 340, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([17, -18, 19], 20, 17, 17, 0, 17, [289, -306, 323], [289, -306, 323], Yes, 340, 340, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
68. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[18, -19, 20], objective coefficient=21, unrestricted value=-18. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([18, -19, 20], 21, -18, 0, 18, -18, [-324, 342, -360], [-324, 342, -360], Yes, -378, -378, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([18, -19, 20], 21, -18, 0, 18, -18, [-324, 342, -360], [-324, 342, -360], Yes, -378, -378, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
69. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[19, -20, 21], objective coefficient=22, unrestricted value=19. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([19, -20, 21], 22, 19, 19, 0, 19, [361, -380, 399], [361, -380, 399], Yes, 418, 418, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([19, -20, 21], 22, 19, 19, 0, 19, [361, -380, 399], [361, -380, 399], Yes, 418, 418, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
70. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[20, -21, 22], objective coefficient=23, unrestricted value=-20. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([20, -21, 22], 23, -20, 0, 20, -20, [-400, 420, -440], [-400, 420, -440], Yes, -460, -460, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([20, -21, 22], 23, -20, 0, 20, -20, [-400, 420, -440], [-400, 420, -440], Yes, -460, -460, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
71. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[21, -22, 23], objective coefficient=24, unrestricted value=21. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([21, -22, 23], 24, 21, 21, 0, 21, [441, -462, 483], [441, -462, 483], Yes, 504, 504, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([21, -22, 23], 24, 21, 21, 0, 21, [441, -462, 483], [441, -462, 483], Yes, 504, 504, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
72. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[22, -23, 24], objective coefficient=25, unrestricted value=-22. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([22, -23, 24], 25, -22, 0, 22, -22, [-484, 506, -528], [-484, 506, -528], Yes, -550, -550, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([22, -23, 24], 25, -22, 0, 22, -22, [-484, 506, -528], [-484, 506, -528], Yes, -550, -550, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
73. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[23, -24, 25], objective coefficient=26, unrestricted value=23. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([23, -24, 25], 26, 23, 23, 0, 23, [529, -552, 575], [529, -552, 575], Yes, 598, 598, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([23, -24, 25], 26, 23, 23, 0, 23, [529, -552, 575], [529, -552, 575], Yes, 598, 598, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
74. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[24, -25, 26], objective coefficient=27, unrestricted value=-24. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([24, -25, 26], 27, -24, 0, 24, -24, [-576, 600, -624], [-576, 600, -624], Yes, -648, -648, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([24, -25, 26], 27, -24, 0, 24, -24, [-576, 600, -624], [-576, 600, -624], Yes, -648, -648, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
75. A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split
Use column=[25, -26, 27], objective coefficient=28, unrestricted value=25. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).
This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations.
View answer and explanation
([25, -26, 27], 28, 25, 25, 0, 25, [625, -650, 675], [625, -650, 675], Yes, 700, 700, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)
Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([25, -26, 27], 28, 25, 25, 0, 25, [625, -650, 675], [625, -650, 675], Yes, 700, 700, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-unrestricted-split · candidate
76. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=greater_equal, coefficients=[1, -2, 1], rhs=3. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(greater_equal, [1, -2, 1], 3, [-1, 2, -1], -3, multiply_constraint_by_minus_one_to_obtain_less_equal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [1, -2, 1], 3, [-1, 2, -1], -3, multiply_constraint_by_minus_one_to_obtain_less_equal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
77. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=equality, coefficients=[2, -3, 1], rhs=4. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(equality, [2, -3, 1], 4, [[2, -3, 1], 4], [[-2, 3, -1], -4], replace_equality_by_two_opposite_less_equal_constraints)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [2, -3, 1], 4, [[2, -3, 1], 4], [[-2, 3, -1], -4], replace_equality_by_two_opposite_less_equal_constraints).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
78. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=minimize, coefficients=[3, -4, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(minimize, [3, -4, 1], [-3, 4, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [3, -4, 1], [-3, 4, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
79. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=greater_equal, coefficients=[4, -5, 1], rhs=6. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(greater_equal, [4, -5, 1], 6, [-4, 5, -1], -6, multiply_constraint_by_minus_one_to_obtain_less_equal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [4, -5, 1], 6, [-4, 5, -1], -6, multiply_constraint_by_minus_one_to_obtain_less_equal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
80. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=equality, coefficients=[5, -6, 1], rhs=7. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(equality, [5, -6, 1], 7, [[5, -6, 1], 7], [[-5, 6, -1], -7], replace_equality_by_two_opposite_less_equal_constraints)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [5, -6, 1], 7, [[5, -6, 1], 7], [[-5, 6, -1], -7], replace_equality_by_two_opposite_less_equal_constraints).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
81. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=minimize, coefficients=[6, -7, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(minimize, [6, -7, 1], [-6, 7, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [6, -7, 1], [-6, 7, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
82. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=greater_equal, coefficients=[7, -8, 1], rhs=9. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(greater_equal, [7, -8, 1], 9, [-7, 8, -1], -9, multiply_constraint_by_minus_one_to_obtain_less_equal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [7, -8, 1], 9, [-7, 8, -1], -9, multiply_constraint_by_minus_one_to_obtain_less_equal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
83. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=equality, coefficients=[8, -9, 1], rhs=10. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(equality, [8, -9, 1], 10, [[8, -9, 1], 10], [[-8, 9, -1], -10], replace_equality_by_two_opposite_less_equal_constraints)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [8, -9, 1], 10, [[8, -9, 1], 10], [[-8, 9, -1], -10], replace_equality_by_two_opposite_less_equal_constraints).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
84. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=minimize, coefficients=[9, -10, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(minimize, [9, -10, 1], [-9, 10, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [9, -10, 1], [-9, 10, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
85. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=greater_equal, coefficients=[10, -11, 1], rhs=12. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(greater_equal, [10, -11, 1], 12, [-10, 11, -1], -12, multiply_constraint_by_minus_one_to_obtain_less_equal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [10, -11, 1], 12, [-10, 11, -1], -12, multiply_constraint_by_minus_one_to_obtain_less_equal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
86. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=equality, coefficients=[11, -12, 1], rhs=13. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(equality, [11, -12, 1], 13, [[11, -12, 1], 13], [[-11, 12, -1], -13], replace_equality_by_two_opposite_less_equal_constraints)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [11, -12, 1], 13, [[11, -12, 1], 13], [[-11, 12, -1], -13], replace_equality_by_two_opposite_less_equal_constraints).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
87. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=minimize, coefficients=[12, -13, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(minimize, [12, -13, 1], [-12, 13, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [12, -13, 1], [-12, 13, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
88. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=greater_equal, coefficients=[13, -14, 1], rhs=15. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(greater_equal, [13, -14, 1], 15, [-13, 14, -1], -15, multiply_constraint_by_minus_one_to_obtain_less_equal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [13, -14, 1], 15, [-13, 14, -1], -15, multiply_constraint_by_minus_one_to_obtain_less_equal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
89. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=equality, coefficients=[14, -15, 1], rhs=16. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(equality, [14, -15, 1], 16, [[14, -15, 1], 16], [[-14, 15, -1], -16], replace_equality_by_two_opposite_less_equal_constraints)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [14, -15, 1], 16, [[14, -15, 1], 16], [[-14, 15, -1], -16], replace_equality_by_two_opposite_less_equal_constraints).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
90. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=minimize, coefficients=[15, -16, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(minimize, [15, -16, 1], [-15, 16, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [15, -16, 1], [-15, 16, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
91. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=greater_equal, coefficients=[16, -17, 1], rhs=18. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(greater_equal, [16, -17, 1], 18, [-16, 17, -1], -18, multiply_constraint_by_minus_one_to_obtain_less_equal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [16, -17, 1], 18, [-16, 17, -1], -18, multiply_constraint_by_minus_one_to_obtain_less_equal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
92. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=equality, coefficients=[17, -18, 1], rhs=19. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(equality, [17, -18, 1], 19, [[17, -18, 1], 19], [[-17, 18, -1], -19], replace_equality_by_two_opposite_less_equal_constraints)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [17, -18, 1], 19, [[17, -18, 1], 19], [[-17, 18, -1], -19], replace_equality_by_two_opposite_less_equal_constraints).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
93. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=minimize, coefficients=[18, -19, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(minimize, [18, -19, 1], [-18, 19, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [18, -19, 1], [-18, 19, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
94. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=greater_equal, coefficients=[19, -20, 1], rhs=21. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(greater_equal, [19, -20, 1], 21, [-19, 20, -1], -21, multiply_constraint_by_minus_one_to_obtain_less_equal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [19, -20, 1], 21, [-19, 20, -1], -21, multiply_constraint_by_minus_one_to_obtain_less_equal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
95. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=equality, coefficients=[20, -21, 1], rhs=22. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(equality, [20, -21, 1], 22, [[20, -21, 1], 22], [[-20, 21, -1], -22], replace_equality_by_two_opposite_less_equal_constraints)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [20, -21, 1], 22, [[20, -21, 1], 22], [[-20, 21, -1], -22], replace_equality_by_two_opposite_less_equal_constraints).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
96. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=minimize, coefficients=[21, -22, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(minimize, [21, -22, 1], [-21, 22, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [21, -22, 1], [-21, 22, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
97. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=greater_equal, coefficients=[22, -23, 1], rhs=24. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(greater_equal, [22, -23, 1], 24, [-22, 23, -1], -24, multiply_constraint_by_minus_one_to_obtain_less_equal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [22, -23, 1], 24, [-22, 23, -1], -24, multiply_constraint_by_minus_one_to_obtain_less_equal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
98. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=equality, coefficients=[23, -24, 1], rhs=25. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(equality, [23, -24, 1], 25, [[23, -24, 1], 25], [[-23, 24, -1], -25], replace_equality_by_two_opposite_less_equal_constraints)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [23, -24, 1], 25, [[23, -24, 1], 25], [[-23, 24, -1], -25], replace_equality_by_two_opposite_less_equal_constraints).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
99. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=minimize, coefficients=[24, -25, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(minimize, [24, -25, 1], [-24, 25, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [24, -25, 1], [-24, 25, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
100. Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles
Use mode=greater_equal, coefficients=[25, -26, 1], rhs=27. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).
All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality.
View answer and explanation
(greater_equal, [25, -26, 1], 27, [-25, 26, -1], -27, multiply_constraint_by_minus_one_to_obtain_less_equal)
For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [25, -26, 1], 27, [-25, 26, -1], -27, multiply_constraint_by_minus_one_to_obtain_less_equal).
Reviewed step-by-step derivation
Standard form of a linear program · qf-mod45-lp-standard-convention-boundary · candidate
Canonical question JSONL · English JSONL · Validation report