Fixed Dual Conventions, Feasible Weak Chains and Zero-Gap Certificates
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batch-20261015-advance-20261115-r04-linear-program-duality-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. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[2, 3], c=[1, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [2, 3], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [2, 3], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
2. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[3, 4], c=[2, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [3, 4], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [3, 4], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
3. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[4, 5], c=[3, 3]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [4, 5], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [4, 5], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
4. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[5, 6], c=[4, 4]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [5, 6], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [5, 6], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
5. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[6, 7], c=[1, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [6, 7], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [6, 7], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
6. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[7, 8], c=[2, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [7, 8], [2, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [7, 8], [2, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
7. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[8, 9], c=[3, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [8, 9], [3, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [8, 9], [3, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
8. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[9, 10], c=[4, 3]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [9, 10], [4, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [9, 10], [4, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
9. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[10, 11], c=[1, 4]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [10, 11], [1, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [10, 11], [1, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
10. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[11, 12], c=[2, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [11, 12], [2, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [11, 12], [2, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
11. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[12, 13], c=[3, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [12, 13], [3, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [12, 13], [3, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
12. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[13, 14], c=[4, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [13, 14], [4, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [13, 14], [4, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
13. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[14, 15], c=[1, 3]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [14, 15], [1, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [14, 15], [1, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
14. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[15, 16], c=[2, 4]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [15, 16], [2, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [15, 16], [2, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
15. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[16, 17], c=[3, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [16, 17], [3, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [16, 17], [3, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
16. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[17, 18], c=[4, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [17, 18], [4, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [17, 18], [4, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
17. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[18, 19], c=[1, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [18, 19], [1, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [18, 19], [1, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
18. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[19, 20], c=[2, 3]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [19, 20], [2, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [19, 20], [2, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
19. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[20, 21], c=[3, 4]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [20, 21], [3, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [20, 21], [3, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
20. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[21, 22], c=[4, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [21, 22], [4, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [21, 22], [4, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
21. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[22, 23], c=[1, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [22, 23], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [22, 23], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
22. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[23, 24], c=[2, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [23, 24], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [23, 24], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
23. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[24, 25], c=[3, 3]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [24, 25], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [24, 25], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
24. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[25, 26], c=[4, 4]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [25, 26], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [25, 26], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
25. Construct the Dual from One Fixed Primal Convention and the Exact Transpose
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[26, 27], c=[1, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).
View answer and explanation
(2, 2, [[1, 0], [0, 1]], [26, 27], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)
Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [26, 27], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-construction-qualification · candidate
26. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[2, 3], c=[1, 1]. Candidate x=[2/2, 3/2], y=[2, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([1, 3/2], [2, 2], [1, 3/2], [2, 2], Yes, Yes, 5/2, 5, 10, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([1, 3/2], [2, 2], [1, 3/2], [2, 2], Yes, Yes, 5/2, 5, 10, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
27. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[3, 4], c=[2, 2]. Candidate x=[3/2, 4/2], y=[3, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([3/2, 2], [3, 3], [3/2, 2], [3, 3], Yes, Yes, 7, 21/2, 21, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([3/2, 2], [3, 3], [3/2, 2], [3, 3], Yes, Yes, 7, 21/2, 21, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
28. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[4, 5], c=[3, 3]. Candidate x=[4/2, 5/2], y=[4, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([2, 5/2], [4, 4], [2, 5/2], [4, 4], Yes, Yes, 27/2, 18, 36, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([2, 5/2], [4, 4], [2, 5/2], [4, 4], Yes, Yes, 27/2, 18, 36, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
29. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[5, 6], c=[4, 4]. Candidate x=[5/2, 6/2], y=[5, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([5/2, 3], [5, 5], [5/2, 3], [5, 5], Yes, Yes, 22, 55/2, 55, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([5/2, 3], [5, 5], [5/2, 3], [5, 5], Yes, Yes, 22, 55/2, 55, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
30. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[6, 7], c=[1, 5]. Candidate x=[6/2, 7/2], y=[2, 6]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([3, 7/2], [2, 6], [3, 7/2], [2, 6], Yes, Yes, 41/2, 27, 54, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([3, 7/2], [2, 6], [3, 7/2], [2, 6], Yes, Yes, 41/2, 27, 54, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
31. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[7, 8], c=[2, 1]. Candidate x=[7/2, 8/2], y=[3, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([7/2, 4], [3, 2], [7/2, 4], [3, 2], Yes, Yes, 11, 37/2, 37, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([7/2, 4], [3, 2], [7/2, 4], [3, 2], Yes, Yes, 11, 37/2, 37, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
32. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[8, 9], c=[3, 2]. Candidate x=[8/2, 9/2], y=[4, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([4, 9/2], [4, 3], [4, 9/2], [4, 3], Yes, Yes, 21, 59/2, 59, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([4, 9/2], [4, 3], [4, 9/2], [4, 3], Yes, Yes, 21, 59/2, 59, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
33. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[9, 10], c=[4, 3]. Candidate x=[9/2, 10/2], y=[5, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([9/2, 5], [5, 4], [9/2, 5], [5, 4], Yes, Yes, 33, 85/2, 85, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([9/2, 5], [5, 4], [9/2, 5], [5, 4], Yes, Yes, 33, 85/2, 85, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
34. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[10, 11], c=[1, 4]. Candidate x=[10/2, 11/2], y=[2, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([5, 11/2], [2, 5], [5, 11/2], [2, 5], Yes, Yes, 27, 75/2, 75, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([5, 11/2], [2, 5], [5, 11/2], [2, 5], Yes, Yes, 27, 75/2, 75, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
35. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[11, 12], c=[2, 5]. Candidate x=[11/2, 12/2], y=[3, 6]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([11/2, 6], [3, 6], [11/2, 6], [3, 6], Yes, Yes, 41, 105/2, 105, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([11/2, 6], [3, 6], [11/2, 6], [3, 6], Yes, Yes, 41, 105/2, 105, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
36. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[12, 13], c=[3, 1]. Candidate x=[12/2, 13/2], y=[4, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([6, 13/2], [4, 2], [6, 13/2], [4, 2], Yes, Yes, 49/2, 37, 74, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([6, 13/2], [4, 2], [6, 13/2], [4, 2], Yes, Yes, 49/2, 37, 74, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
37. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[13, 14], c=[4, 2]. Candidate x=[13/2, 14/2], y=[5, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([13/2, 7], [5, 3], [13/2, 7], [5, 3], Yes, Yes, 40, 107/2, 107, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([13/2, 7], [5, 3], [13/2, 7], [5, 3], Yes, Yes, 40, 107/2, 107, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
38. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[14, 15], c=[1, 3]. Candidate x=[14/2, 15/2], y=[2, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([7, 15/2], [2, 4], [7, 15/2], [2, 4], Yes, Yes, 59/2, 44, 88, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([7, 15/2], [2, 4], [7, 15/2], [2, 4], Yes, Yes, 59/2, 44, 88, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
39. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[15, 16], c=[2, 4]. Candidate x=[15/2, 16/2], y=[3, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([15/2, 8], [3, 5], [15/2, 8], [3, 5], Yes, Yes, 47, 125/2, 125, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([15/2, 8], [3, 5], [15/2, 8], [3, 5], Yes, Yes, 47, 125/2, 125, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
40. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[16, 17], c=[3, 5]. Candidate x=[16/2, 17/2], y=[4, 6]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([8, 17/2], [4, 6], [8, 17/2], [4, 6], Yes, Yes, 133/2, 83, 166, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([8, 17/2], [4, 6], [8, 17/2], [4, 6], Yes, Yes, 133/2, 83, 166, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
41. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[17, 18], c=[4, 1]. Candidate x=[17/2, 18/2], y=[5, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([17/2, 9], [5, 2], [17/2, 9], [5, 2], Yes, Yes, 43, 121/2, 121, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([17/2, 9], [5, 2], [17/2, 9], [5, 2], Yes, Yes, 43, 121/2, 121, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
42. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[18, 19], c=[1, 2]. Candidate x=[18/2, 19/2], y=[2, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([9, 19/2], [2, 3], [9, 19/2], [2, 3], Yes, Yes, 28, 93/2, 93, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([9, 19/2], [2, 3], [9, 19/2], [2, 3], Yes, Yes, 28, 93/2, 93, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
43. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[19, 20], c=[2, 3]. Candidate x=[19/2, 20/2], y=[3, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([19/2, 10], [3, 4], [19/2, 10], [3, 4], Yes, Yes, 49, 137/2, 137, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([19/2, 10], [3, 4], [19/2, 10], [3, 4], Yes, Yes, 49, 137/2, 137, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
44. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[20, 21], c=[3, 4]. Candidate x=[20/2, 21/2], y=[4, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([10, 21/2], [4, 5], [10, 21/2], [4, 5], Yes, Yes, 72, 185/2, 185, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([10, 21/2], [4, 5], [10, 21/2], [4, 5], Yes, Yes, 72, 185/2, 185, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
45. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[21, 22], c=[4, 5]. Candidate x=[21/2, 22/2], y=[5, 6]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([21/2, 11], [5, 6], [21/2, 11], [5, 6], Yes, Yes, 97, 237/2, 237, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([21/2, 11], [5, 6], [21/2, 11], [5, 6], Yes, Yes, 97, 237/2, 237, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
46. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[22, 23], c=[1, 1]. Candidate x=[22/2, 23/2], y=[2, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([11, 23/2], [2, 2], [11, 23/2], [2, 2], Yes, Yes, 45/2, 45, 90, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([11, 23/2], [2, 2], [11, 23/2], [2, 2], Yes, Yes, 45/2, 45, 90, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
47. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[23, 24], c=[2, 2]. Candidate x=[23/2, 24/2], y=[3, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([23/2, 12], [3, 3], [23/2, 12], [3, 3], Yes, Yes, 47, 141/2, 141, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([23/2, 12], [3, 3], [23/2, 12], [3, 3], Yes, Yes, 47, 141/2, 141, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
48. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[24, 25], c=[3, 3]. Candidate x=[24/2, 25/2], y=[4, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([12, 25/2], [4, 4], [12, 25/2], [4, 4], Yes, Yes, 147/2, 98, 196, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([12, 25/2], [4, 4], [12, 25/2], [4, 4], Yes, Yes, 147/2, 98, 196, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
49. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[25, 26], c=[4, 4]. Candidate x=[25/2, 26/2], y=[5, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([25/2, 13], [5, 5], [25/2, 13], [5, 5], Yes, Yes, 102, 255/2, 255, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([25/2, 13], [5, 5], [25/2, 13], [5, 5], Yes, Yes, 102, 255/2, 255, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
50. Both Feasibility Checks Precede the Weak-Duality Inequality Chain
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[26, 27], c=[1, 5]. Candidate x=[26/2, 27/2], y=[2, 6]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).
View answer and explanation
([13, 27/2], [2, 6], [13, 27/2], [2, 6], Yes, Yes, 161/2, 107, 214, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)
Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([13, 27/2], [2, 6], [13, 27/2], [2, 6], Yes, Yes, 161/2, 107, 214, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-weak-duality-chain · candidate
51. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[2, 3], c=[1, 1]. Candidate x=[2, 3], y=[1, 1]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([2, 3], [1, 1], [2, 3], [1, 1], Yes, Yes, 5, 5, 5, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([2, 3], [1, 1], [2, 3], [1, 1], Yes, Yes, 5, 5, 5, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
52. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[3, 4], c=[2, 2]. Candidate x=[3, 4], y=[2, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([3, 4], [2, 2], [3, 4], [2, 2], Yes, Yes, 14, 14, 14, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([3, 4], [2, 2], [3, 4], [2, 2], Yes, Yes, 14, 14, 14, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
53. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[4, 5], c=[3, 3]. Candidate x=[4, 5], y=[3, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([4, 5], [3, 3], [4, 5], [3, 3], Yes, Yes, 27, 27, 27, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([4, 5], [3, 3], [4, 5], [3, 3], Yes, Yes, 27, 27, 27, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
54. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[5, 6], c=[4, 4]. Candidate x=[5, 6], y=[4, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([5, 6], [4, 4], [5, 6], [4, 4], Yes, Yes, 44, 44, 44, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([5, 6], [4, 4], [5, 6], [4, 4], Yes, Yes, 44, 44, 44, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
55. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[6, 7], c=[1, 5]. Candidate x=[6, 7], y=[1, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([6, 7], [1, 5], [6, 7], [1, 5], Yes, Yes, 41, 41, 41, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([6, 7], [1, 5], [6, 7], [1, 5], Yes, Yes, 41, 41, 41, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
56. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[7, 8], c=[2, 1]. Candidate x=[7, 8], y=[2, 1]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([7, 8], [2, 1], [7, 8], [2, 1], Yes, Yes, 22, 22, 22, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([7, 8], [2, 1], [7, 8], [2, 1], Yes, Yes, 22, 22, 22, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
57. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[8, 9], c=[3, 2]. Candidate x=[8, 9], y=[3, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([8, 9], [3, 2], [8, 9], [3, 2], Yes, Yes, 42, 42, 42, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([8, 9], [3, 2], [8, 9], [3, 2], Yes, Yes, 42, 42, 42, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
58. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[9, 10], c=[4, 3]. Candidate x=[9, 10], y=[4, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([9, 10], [4, 3], [9, 10], [4, 3], Yes, Yes, 66, 66, 66, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([9, 10], [4, 3], [9, 10], [4, 3], Yes, Yes, 66, 66, 66, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
59. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[10, 11], c=[1, 4]. Candidate x=[10, 11], y=[1, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([10, 11], [1, 4], [10, 11], [1, 4], Yes, Yes, 54, 54, 54, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([10, 11], [1, 4], [10, 11], [1, 4], Yes, Yes, 54, 54, 54, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
60. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[11, 12], c=[2, 5]. Candidate x=[11, 12], y=[2, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([11, 12], [2, 5], [11, 12], [2, 5], Yes, Yes, 82, 82, 82, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([11, 12], [2, 5], [11, 12], [2, 5], Yes, Yes, 82, 82, 82, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
61. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[12, 13], c=[3, 1]. Candidate x=[12, 13], y=[3, 1]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([12, 13], [3, 1], [12, 13], [3, 1], Yes, Yes, 49, 49, 49, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([12, 13], [3, 1], [12, 13], [3, 1], Yes, Yes, 49, 49, 49, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
62. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[13, 14], c=[4, 2]. Candidate x=[13, 14], y=[4, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([13, 14], [4, 2], [13, 14], [4, 2], Yes, Yes, 80, 80, 80, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([13, 14], [4, 2], [13, 14], [4, 2], Yes, Yes, 80, 80, 80, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
63. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[14, 15], c=[1, 3]. Candidate x=[14, 15], y=[1, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([14, 15], [1, 3], [14, 15], [1, 3], Yes, Yes, 59, 59, 59, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([14, 15], [1, 3], [14, 15], [1, 3], Yes, Yes, 59, 59, 59, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
64. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[15, 16], c=[2, 4]. Candidate x=[15, 16], y=[2, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([15, 16], [2, 4], [15, 16], [2, 4], Yes, Yes, 94, 94, 94, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([15, 16], [2, 4], [15, 16], [2, 4], Yes, Yes, 94, 94, 94, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
65. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[16, 17], c=[3, 5]. Candidate x=[16, 17], y=[3, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([16, 17], [3, 5], [16, 17], [3, 5], Yes, Yes, 133, 133, 133, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([16, 17], [3, 5], [16, 17], [3, 5], Yes, Yes, 133, 133, 133, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
66. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[17, 18], c=[4, 1]. Candidate x=[17, 18], y=[4, 1]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([17, 18], [4, 1], [17, 18], [4, 1], Yes, Yes, 86, 86, 86, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([17, 18], [4, 1], [17, 18], [4, 1], Yes, Yes, 86, 86, 86, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
67. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[18, 19], c=[1, 2]. Candidate x=[18, 19], y=[1, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([18, 19], [1, 2], [18, 19], [1, 2], Yes, Yes, 56, 56, 56, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([18, 19], [1, 2], [18, 19], [1, 2], Yes, Yes, 56, 56, 56, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
68. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[19, 20], c=[2, 3]. Candidate x=[19, 20], y=[2, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([19, 20], [2, 3], [19, 20], [2, 3], Yes, Yes, 98, 98, 98, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([19, 20], [2, 3], [19, 20], [2, 3], Yes, Yes, 98, 98, 98, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
69. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[20, 21], c=[3, 4]. Candidate x=[20, 21], y=[3, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([20, 21], [3, 4], [20, 21], [3, 4], Yes, Yes, 144, 144, 144, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([20, 21], [3, 4], [20, 21], [3, 4], Yes, Yes, 144, 144, 144, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
70. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[21, 22], c=[4, 5]. Candidate x=[21, 22], y=[4, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([21, 22], [4, 5], [21, 22], [4, 5], Yes, Yes, 194, 194, 194, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([21, 22], [4, 5], [21, 22], [4, 5], Yes, Yes, 194, 194, 194, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
71. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[22, 23], c=[1, 1]. Candidate x=[22, 23], y=[1, 1]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([22, 23], [1, 1], [22, 23], [1, 1], Yes, Yes, 45, 45, 45, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([22, 23], [1, 1], [22, 23], [1, 1], Yes, Yes, 45, 45, 45, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
72. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[23, 24], c=[2, 2]. Candidate x=[23, 24], y=[2, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([23, 24], [2, 2], [23, 24], [2, 2], Yes, Yes, 94, 94, 94, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([23, 24], [2, 2], [23, 24], [2, 2], Yes, Yes, 94, 94, 94, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
73. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[24, 25], c=[3, 3]. Candidate x=[24, 25], y=[3, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([24, 25], [3, 3], [24, 25], [3, 3], Yes, Yes, 147, 147, 147, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([24, 25], [3, 3], [24, 25], [3, 3], Yes, Yes, 147, 147, 147, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
74. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[25, 26], c=[4, 4]. Candidate x=[25, 26], y=[4, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([25, 26], [4, 4], [25, 26], [4, 4], Yes, Yes, 204, 204, 204, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([25, 26], [4, 4], [25, 26], [4, 4], Yes, Yes, 204, 204, 204, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
75. A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[26, 27], c=[1, 5]. Candidate x=[26, 27], y=[1, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).
View answer and explanation
([26, 27], [1, 5], [26, 27], [1, 5], Yes, Yes, 161, 161, 161, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)
Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([26, 27], [1, 5], [26, 27], [1, 5], Yes, Yes, 161, 161, 161, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-zero-gap-certificate · candidate
76. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[2], c=[1]. Candidate x=[3], y=[3/2]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(primal_infeasible_equal_objective, [3], [3/2], [3], [3/2], No, Yes, 3, 9/2, 3, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [3], [3/2], [3], [3/2], No, Yes, 3, 9/2, 3, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
77. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[3]. Candidate x=[1/3], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(dual_infeasible_equal_objective, [1/3], [1], [1/3], [1], Yes, No, 1, 1/3, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/3], [1], [1/3], [1], Yes, No, 1, 1/3, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
78. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[4], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 8, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 8, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
79. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[5], c=[1]. Candidate x=[6], y=[6/5]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(primal_infeasible_equal_objective, [6], [6/5], [6], [6/5], No, Yes, 6, 36/5, 6, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [6], [6/5], [6], [6/5], No, Yes, 6, 36/5, 6, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
80. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[6]. Candidate x=[1/6], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(dual_infeasible_equal_objective, [1/6], [1], [1/6], [1], Yes, No, 1, 1/6, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/6], [1], [1/6], [1], Yes, No, 1, 1/6, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
81. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[7], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 14, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 14, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
82. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[8], c=[1]. Candidate x=[9], y=[9/8]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(primal_infeasible_equal_objective, [9], [9/8], [9], [9/8], No, Yes, 9, 81/8, 9, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [9], [9/8], [9], [9/8], No, Yes, 9, 81/8, 9, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
83. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[9]. Candidate x=[1/9], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(dual_infeasible_equal_objective, [1/9], [1], [1/9], [1], Yes, No, 1, 1/9, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/9], [1], [1/9], [1], Yes, No, 1, 1/9, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
84. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[10], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 20, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 20, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
85. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[11], c=[1]. Candidate x=[12], y=[12/11]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(primal_infeasible_equal_objective, [12], [12/11], [12], [12/11], No, Yes, 12, 144/11, 12, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [12], [12/11], [12], [12/11], No, Yes, 12, 144/11, 12, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
86. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[12]. Candidate x=[1/12], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(dual_infeasible_equal_objective, [1/12], [1], [1/12], [1], Yes, No, 1, 1/12, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/12], [1], [1/12], [1], Yes, No, 1, 1/12, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
87. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[13], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 26, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 26, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
88. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[14], c=[1]. Candidate x=[15], y=[15/14]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(primal_infeasible_equal_objective, [15], [15/14], [15], [15/14], No, Yes, 15, 225/14, 15, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [15], [15/14], [15], [15/14], No, Yes, 15, 225/14, 15, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
89. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[15]. Candidate x=[1/15], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(dual_infeasible_equal_objective, [1/15], [1], [1/15], [1], Yes, No, 1, 1/15, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/15], [1], [1/15], [1], Yes, No, 1, 1/15, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
90. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[16], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 32, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 32, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
91. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[17], c=[1]. Candidate x=[18], y=[18/17]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(primal_infeasible_equal_objective, [18], [18/17], [18], [18/17], No, Yes, 18, 324/17, 18, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [18], [18/17], [18], [18/17], No, Yes, 18, 324/17, 18, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
92. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[18]. Candidate x=[1/18], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(dual_infeasible_equal_objective, [1/18], [1], [1/18], [1], Yes, No, 1, 1/18, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/18], [1], [1/18], [1], Yes, No, 1, 1/18, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
93. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[19], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 38, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 38, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
94. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[20], c=[1]. Candidate x=[21], y=[21/20]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(primal_infeasible_equal_objective, [21], [21/20], [21], [21/20], No, Yes, 21, 441/20, 21, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [21], [21/20], [21], [21/20], No, Yes, 21, 441/20, 21, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
95. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[21]. Candidate x=[1/21], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(dual_infeasible_equal_objective, [1/21], [1], [1/21], [1], Yes, No, 1, 1/21, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/21], [1], [1/21], [1], Yes, No, 1, 1/21, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
96. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[22], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 44, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 44, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
97. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[23], c=[1]. Candidate x=[24], y=[24/23]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(primal_infeasible_equal_objective, [24], [24/23], [24], [24/23], No, Yes, 24, 576/23, 24, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [24], [24/23], [24], [24/23], No, Yes, 24, 576/23, 24, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
98. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[24]. Candidate x=[1/24], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(dual_infeasible_equal_objective, [1/24], [1], [1/24], [1], Yes, No, 1, 1/24, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/24], [1], [1/24], [1], Yes, No, 1, 1/24, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
99. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[25], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 50, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 50, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
100. Equal Objective Values and Finite Grids Do Not Replace Theorem Premises
Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[26], c=[1]. Candidate x=[27], y=[27/26]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).
All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair.
View answer and explanation
(primal_infeasible_equal_objective, [27], [27/26], [27], [27/26], No, Yes, 27, 729/26, 27, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)
Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [27], [27/26], [27], [27/26], No, Yes, 27, 729/26, 27, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases).
Reviewed step-by-step derivation
Linear-programming duality · qf-mod46-lpdual-strong-boundary · candidate
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