Convex-Problem Qualification, Feasibility, Evaluation and Pairwise Comparison
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261025-r03-convex-optimization-problem-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -12, -3], not ascending-power order. The declared interval [lower,upper]=[-2, 3] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-1, -12, -3], [-2, 3], No, Yes, No, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -12, -3], [-2, 3], No, Yes, No, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
2. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -11, -2], not ascending-power order. The declared interval [lower,upper]=[-3, 4] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, -11, -2], [-3, 4], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -11, -2], [-3, 4], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
3. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-4, 5] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, -10, -1], [-4, 5], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -10, -1], [-4, 5], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
4. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -9, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 6] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, -9, 0], [-5, 6], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -9, 0], [-5, 6], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
5. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -8, 1], not ascending-power order. The declared interval [lower,upper]=[-6, 7] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, -8, 1], [-6, 7], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -8, 1], [-6, 7], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
6. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -7, 2], not ascending-power order. The declared interval [lower,upper]=[-7, 8] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-1, -7, 2], [-7, 8], No, Yes, No, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -7, 2], [-7, 8], No, Yes, No, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
7. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -6, 3], not ascending-power order. The declared interval [lower,upper]=[-8, 9] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, -6, 3], [-8, 9], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -6, 3], [-8, 9], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
8. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -5, -3], not ascending-power order. The declared interval [lower,upper]=[-9, 10] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, -5, -3], [-9, 10], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -5, -3], [-9, 10], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
9. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -4, -2], not ascending-power order. The declared interval [lower,upper]=[-10, 11] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, -4, -2], [-10, 11], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -4, -2], [-10, 11], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
10. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -3, -1], not ascending-power order. The declared interval [lower,upper]=[-11, 12] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, -3, -1], [-11, 12], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -3, -1], [-11, 12], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
11. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -2, 0], not ascending-power order. The declared interval [lower,upper]=[-12, 13] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-1, -2, 0], [-12, 13], No, Yes, No, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -2, 0], [-12, 13], No, Yes, No, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
12. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -1, 1], not ascending-power order. The declared interval [lower,upper]=[-13, 14] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, -1, 1], [-13, 14], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -1, 1], [-13, 14], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
13. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 0, 2], not ascending-power order. The declared interval [lower,upper]=[-14, 15] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 0, 2], [-14, 15], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2], [-14, 15], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
14. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 1, 3], not ascending-power order. The declared interval [lower,upper]=[-15, 16] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, 1, 3], [-15, 16], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3], [-15, 16], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
15. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 2, -3], not ascending-power order. The declared interval [lower,upper]=[-16, 17] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, 2, -3], [-16, 17], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3], [-16, 17], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
16. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 3, -2], not ascending-power order. The declared interval [lower,upper]=[-17, 18] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-1, 3, -2], [-17, 18], No, Yes, No, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 3, -2], [-17, 18], No, Yes, No, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
17. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 4, -1], not ascending-power order. The declared interval [lower,upper]=[-18, 19] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 4, -1], [-18, 19], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 4, -1], [-18, 19], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
18. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 5, 0], not ascending-power order. The declared interval [lower,upper]=[-19, 20] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, 5, 0], [-19, 20], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 5, 0], [-19, 20], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
19. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 6, 1], not ascending-power order. The declared interval [lower,upper]=[-20, 21] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, 6, 1], [-20, 21], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 6, 1], [-20, 21], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
20. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 7, 2], not ascending-power order. The declared interval [lower,upper]=[-21, 22] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, 7, 2], [-21, 22], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 7, 2], [-21, 22], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
21. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 8, 3], not ascending-power order. The declared interval [lower,upper]=[-22, 23] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-1, 8, 3], [-22, 23], No, Yes, No, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 8, 3], [-22, 23], No, Yes, No, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
22. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 9, -3], not ascending-power order. The declared interval [lower,upper]=[-23, 24] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, 9, -3], [-23, 24], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 9, -3], [-23, 24], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
23. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-24, 25] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, 10, -2], [-24, 25], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 10, -2], [-24, 25], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
24. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 11, -1], not ascending-power order. The declared interval [lower,upper]=[-25, 26] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, 11, -1], [-25, 26], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 11, -1], [-25, 26], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
25. Separate a Quadratic Objective Certificate from Feasible-Interval Qualification
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 12, 0], not ascending-power order. The declared interval [lower,upper]=[-26, 27] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 12, 0], [-26, 27], Yes, Yes, Yes, qualification_only_no_solution_claim)
Check the two declared qualifications separately. A reversed interval record is outside this nonempty-domain protocol. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 12, 0], [-26, 27], Yes, Yes, Yes, qualification_only_no_solution_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-problem-qualification · candidate
26. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-2, 3], candidate=-3/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-2, 3], -3, No, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 3], -3, No, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
27. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-3, 4], candidate=-2/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-3, 4], -2, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 4], -2, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
28. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-4, 5], candidate=-1/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-4, 5], -1, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 5], -1, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
29. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-5, 6], candidate=0/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-5, 6], 0, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 6], 0, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
30. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-6, 7], candidate=1/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-6, 7], 1, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 7], 1, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
31. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-7, 8], candidate=2/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-7, 8], 2, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 8], 2, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
32. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-8, 9], candidate=3/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-8, 9], 3, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 9], 3, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
33. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-9, 10], candidate=-3/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-9, 10], -3, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 10], -3, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
34. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-10, 11], candidate=-2/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-10, 11], -2, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 11], -2, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
35. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-11, 12], candidate=-1/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-11, 12], -1, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, 12], -1, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
36. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-12, 13], candidate=0/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-12, 13], 0, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, 13], 0, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
37. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-13, 14], candidate=1/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-13, 14], 1, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-13, 14], 1, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
38. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-14, 15], candidate=2/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-14, 15], 2, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-14, 15], 2, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
39. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-15, 16], candidate=3/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-15, 16], 3, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-15, 16], 3, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
40. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-16, 17], candidate=-3/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-16, 17], -3, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-16, 17], -3, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
41. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-17, 18], candidate=-2/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-17, 18], -2, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-17, 18], -2, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
42. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-18, 19], candidate=-1/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-18, 19], -1, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-18, 19], -1, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
43. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-19, 20], candidate=0/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-19, 20], 0, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-19, 20], 0, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
44. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-20, 21], candidate=1/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-20, 21], 1, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-20, 21], 1, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
45. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-21, 22], candidate=2/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-21, 22], 2, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-21, 22], 2, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
46. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-22, 23], candidate=3/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-22, 23], 3, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-22, 23], 3, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
47. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-23, 24], candidate=-3/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-23, 24], -3, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-23, 24], -3, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
48. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-24, 25], candidate=-2/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-24, 25], -2, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-24, 25], -2, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
49. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-25, 26], candidate=-1/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-25, 26], -1, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-25, 26], -1, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
50. Feasibility Alone Supplies No Optimum
Declared nonempty interval record=[-26, 27], candidate=0/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([-26, 27], 0, Yes, feasibility_only)
Compare the exact point with both declared endpoints, retaining equality at a boundary. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-26, 27], 0, Yes, feasibility_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-feasible-point-audit · candidate
51. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -12, -3], not ascending-power order. The declared interval [lower,upper]=[-2, 3] belongs to the function data. point=-3/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, -12, -3], [-2, 3], -3, No, 33, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -12, -3], [-2, 3], -3, No, 33, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
52. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -11, -2], not ascending-power order. The declared interval [lower,upper]=[-3, 4] belongs to the function data. point=-2/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, -11, -2], [-3, 4], -2, Yes, 24, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -11, -2], [-3, 4], -2, Yes, 24, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
53. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-4, 5] belongs to the function data. point=-1/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, -10, -1], [-4, 5], -1, Yes, 11, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -10, -1], [-4, 5], -1, Yes, 11, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
54. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -9, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 6] belongs to the function data. point=0/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, -9, 0], [-5, 6], 0, Yes, 0, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -9, 0], [-5, 6], 0, Yes, 0, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
55. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -8, 1], not ascending-power order. The declared interval [lower,upper]=[-6, 7] belongs to the function data. point=1/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, -8, 1], [-6, 7], 1, Yes, -7, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -8, 1], [-6, 7], 1, Yes, -7, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
56. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -7, 2], not ascending-power order. The declared interval [lower,upper]=[-7, 8] belongs to the function data. point=2/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, -7, 2], [-7, 8], 2, Yes, -8, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -7, 2], [-7, 8], 2, Yes, -8, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
57. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -6, 3], not ascending-power order. The declared interval [lower,upper]=[-8, 9] belongs to the function data. point=3/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, -6, 3], [-8, 9], 3, Yes, 3, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -6, 3], [-8, 9], 3, Yes, 3, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
58. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -5, -3], not ascending-power order. The declared interval [lower,upper]=[-9, 10] belongs to the function data. point=-3/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, -5, -3], [-9, 10], -3, Yes, 39, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -5, -3], [-9, 10], -3, Yes, 39, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
59. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -4, -2], not ascending-power order. The declared interval [lower,upper]=[-10, 11] belongs to the function data. point=-2/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, -4, -2], [-10, 11], -2, Yes, 6, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -4, -2], [-10, 11], -2, Yes, 6, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
60. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -3, -1], not ascending-power order. The declared interval [lower,upper]=[-11, 12] belongs to the function data. point=-1/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, -3, -1], [-11, 12], -1, Yes, 3, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -3, -1], [-11, 12], -1, Yes, 3, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
61. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -2, 0], not ascending-power order. The declared interval [lower,upper]=[-12, 13] belongs to the function data. point=0/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, -2, 0], [-12, 13], 0, Yes, 0, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -2, 0], [-12, 13], 0, Yes, 0, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
62. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -1, 1], not ascending-power order. The declared interval [lower,upper]=[-13, 14] belongs to the function data. point=1/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, -1, 1], [-13, 14], 1, Yes, 3, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -1, 1], [-13, 14], 1, Yes, 3, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
63. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 0, 2], not ascending-power order. The declared interval [lower,upper]=[-14, 15] belongs to the function data. point=2/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 0, 2], [-14, 15], 2, Yes, 2, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2], [-14, 15], 2, Yes, 2, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
64. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 1, 3], not ascending-power order. The declared interval [lower,upper]=[-15, 16] belongs to the function data. point=3/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, 1, 3], [-15, 16], 3, Yes, 15, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3], [-15, 16], 3, Yes, 15, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
65. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 2, -3], not ascending-power order. The declared interval [lower,upper]=[-16, 17] belongs to the function data. point=-3/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, 2, -3], [-16, 17], -3, Yes, 9, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3], [-16, 17], -3, Yes, 9, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
66. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 3, -2], not ascending-power order. The declared interval [lower,upper]=[-17, 18] belongs to the function data. point=-2/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, 3, -2], [-17, 18], -2, Yes, 4, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 3, -2], [-17, 18], -2, Yes, 4, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
67. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 4, -1], not ascending-power order. The declared interval [lower,upper]=[-18, 19] belongs to the function data. point=-1/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 4, -1], [-18, 19], -1, Yes, -5, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 4, -1], [-18, 19], -1, Yes, -5, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
68. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 5, 0], not ascending-power order. The declared interval [lower,upper]=[-19, 20] belongs to the function data. point=0/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, 5, 0], [-19, 20], 0, Yes, 0, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 5, 0], [-19, 20], 0, Yes, 0, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
69. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 6, 1], not ascending-power order. The declared interval [lower,upper]=[-20, 21] belongs to the function data. point=1/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, 6, 1], [-20, 21], 1, Yes, 9, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 6, 1], [-20, 21], 1, Yes, 9, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
70. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 7, 2], not ascending-power order. The declared interval [lower,upper]=[-21, 22] belongs to the function data. point=2/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, 7, 2], [-21, 22], 2, Yes, 28, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 7, 2], [-21, 22], 2, Yes, 28, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
71. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 8, 3], not ascending-power order. The declared interval [lower,upper]=[-22, 23] belongs to the function data. point=3/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 8, 3], [-22, 23], 3, Yes, 27, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 8, 3], [-22, 23], 3, Yes, 27, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
72. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 9, -3], not ascending-power order. The declared interval [lower,upper]=[-23, 24] belongs to the function data. point=-3/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, 9, -3], [-23, 24], -3, Yes, -21, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 9, -3], [-23, 24], -3, Yes, -21, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
73. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-24, 25] belongs to the function data. point=-2/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, 10, -2], [-24, 25], -2, Yes, -14, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 10, -2], [-24, 25], -2, Yes, -14, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
74. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 11, -1], not ascending-power order. The declared interval [lower,upper]=[-25, 26] belongs to the function data. point=-1/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, 11, -1], [-25, 26], -1, Yes, -9, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 11, -1], [-25, 26], -1, Yes, -9, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
75. Evaluate the Polynomial while Separately Reporting Feasibility
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 12, 0], not ascending-power order. The declared interval [lower,upper]=[-26, 27] belongs to the function data. point=0/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 12, 0], [-26, 27], 0, Yes, 0, evaluation_only)
Recompute both the inclusive feasibility predicate and objective value. Do not promote evaluation to optimality. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 12, 0], [-26, 27], 0, Yes, 0, evaluation_only).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-objective-at-feasible-point · candidate
76. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -12, -3], not ascending-power order. The declared interval [lower,upper]=[-2, 3] belongs to the function data. left=-2/1, right=-3/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, -12, -3], [-2, 3], -2, Yes, 21, -3, No, 33, not_comparable, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -12, -3], [-2, 3], -2, Yes, 21, -3, No, 33, not_comparable, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
77. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -11, -2], not ascending-power order. The declared interval [lower,upper]=[-3, 4] belongs to the function data. left=-1/1, right=-2/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, -11, -2], [-3, 4], -1, Yes, 10, -2, Yes, 24, left_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -11, -2], [-3, 4], -1, Yes, 10, -2, Yes, 24, left_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
78. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-4, 5] belongs to the function data. left=0/1, right=-1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, -10, -1], [-4, 5], 0, Yes, -1, -1, Yes, 11, left_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -10, -1], [-4, 5], 0, Yes, -1, -1, Yes, 11, left_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
79. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -9, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 6] belongs to the function data. left=1/1, right=0/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, -9, 0], [-5, 6], 1, Yes, -6, 0, Yes, 0, left_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -9, 0], [-5, 6], 1, Yes, -6, 0, Yes, 0, left_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
80. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -8, 1], not ascending-power order. The declared interval [lower,upper]=[-6, 7] belongs to the function data. left=2/1, right=1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, -8, 1], [-6, 7], 2, Yes, -15, 1, Yes, -7, left_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -8, 1], [-6, 7], 2, Yes, -15, 1, Yes, -7, left_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
81. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -7, 2], not ascending-power order. The declared interval [lower,upper]=[-7, 8] belongs to the function data. left=-2/1, right=2/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, -7, 2], [-7, 8], -2, Yes, 20, 2, Yes, -8, right_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -7, 2], [-7, 8], -2, Yes, 20, 2, Yes, -8, right_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
82. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -6, 3], not ascending-power order. The declared interval [lower,upper]=[-8, 9] belongs to the function data. left=-1/1, right=3/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, -6, 3], [-8, 9], -1, Yes, 11, 3, Yes, 3, right_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -6, 3], [-8, 9], -1, Yes, 11, 3, Yes, 3, right_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
83. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -5, -3], not ascending-power order. The declared interval [lower,upper]=[-9, 10] belongs to the function data. left=0/1, right=-3/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, -5, -3], [-9, 10], 0, Yes, -3, -3, Yes, 39, left_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -5, -3], [-9, 10], 0, Yes, -3, -3, Yes, 39, left_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
84. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -4, -2], not ascending-power order. The declared interval [lower,upper]=[-10, 11] belongs to the function data. left=1/1, right=-2/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, -4, -2], [-10, 11], 1, Yes, -6, -2, Yes, 6, left_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -4, -2], [-10, 11], 1, Yes, -6, -2, Yes, 6, left_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
85. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -3, -1], not ascending-power order. The declared interval [lower,upper]=[-11, 12] belongs to the function data. left=2/1, right=-1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, -3, -1], [-11, 12], 2, Yes, -3, -1, Yes, 3, left_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -3, -1], [-11, 12], 2, Yes, -3, -1, Yes, 3, left_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
86. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -2, 0], not ascending-power order. The declared interval [lower,upper]=[-12, 13] belongs to the function data. left=-2/1, right=0/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, -2, 0], [-12, 13], -2, Yes, 12, 0, Yes, 0, right_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -2, 0], [-12, 13], -2, Yes, 12, 0, Yes, 0, right_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
87. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -1, 1], not ascending-power order. The declared interval [lower,upper]=[-13, 14] belongs to the function data. left=-1/1, right=1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, -1, 1], [-13, 14], -1, Yes, 5, 1, Yes, 3, right_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -1, 1], [-13, 14], -1, Yes, 5, 1, Yes, 3, right_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
88. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 0, 2], not ascending-power order. The declared interval [lower,upper]=[-14, 15] belongs to the function data. left=0/1, right=2/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 0, 2], [-14, 15], 0, Yes, 2, 2, Yes, 2, equal, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2], [-14, 15], 0, Yes, 2, 2, Yes, 2, equal, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
89. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 1, 3], not ascending-power order. The declared interval [lower,upper]=[-15, 16] belongs to the function data. left=1/1, right=3/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, 1, 3], [-15, 16], 1, Yes, 5, 3, Yes, 15, left_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3], [-15, 16], 1, Yes, 5, 3, Yes, 15, left_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
90. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 2, -3], not ascending-power order. The declared interval [lower,upper]=[-16, 17] belongs to the function data. left=2/1, right=-3/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, 2, -3], [-16, 17], 2, Yes, 9, -3, Yes, 9, equal, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3], [-16, 17], 2, Yes, 9, -3, Yes, 9, equal, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
91. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 3, -2], not ascending-power order. The declared interval [lower,upper]=[-17, 18] belongs to the function data. left=-2/1, right=-2/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, 3, -2], [-17, 18], -2, Yes, 4, -2, Yes, 4, equal, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 3, -2], [-17, 18], -2, Yes, 4, -2, Yes, 4, equal, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
92. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 4, -1], not ascending-power order. The declared interval [lower,upper]=[-18, 19] belongs to the function data. left=-1/1, right=-1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 4, -1], [-18, 19], -1, Yes, -5, -1, Yes, -5, equal, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 4, -1], [-18, 19], -1, Yes, -5, -1, Yes, -5, equal, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
93. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 5, 0], not ascending-power order. The declared interval [lower,upper]=[-19, 20] belongs to the function data. left=0/1, right=0/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, 5, 0], [-19, 20], 0, Yes, 0, 0, Yes, 0, equal, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 5, 0], [-19, 20], 0, Yes, 0, 0, Yes, 0, equal, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
94. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 6, 1], not ascending-power order. The declared interval [lower,upper]=[-20, 21] belongs to the function data. left=1/1, right=1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, 6, 1], [-20, 21], 1, Yes, 9, 1, Yes, 9, equal, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 6, 1], [-20, 21], 1, Yes, 9, 1, Yes, 9, equal, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
95. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 7, 2], not ascending-power order. The declared interval [lower,upper]=[-21, 22] belongs to the function data. left=2/1, right=2/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, 7, 2], [-21, 22], 2, Yes, 28, 2, Yes, 28, equal, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 7, 2], [-21, 22], 2, Yes, 28, 2, Yes, 28, equal, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
96. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 8, 3], not ascending-power order. The declared interval [lower,upper]=[-22, 23] belongs to the function data. left=-2/1, right=3/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 8, 3], [-22, 23], -2, Yes, -13, 3, Yes, 27, left_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 8, 3], [-22, 23], -2, Yes, -13, 3, Yes, 27, left_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
97. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 9, -3], not ascending-power order. The declared interval [lower,upper]=[-23, 24] belongs to the function data. left=-1/1, right=-3/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([1, 9, -3], [-23, 24], -1, Yes, -11, -3, Yes, -21, right_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 9, -3], [-23, 24], -1, Yes, -11, -3, Yes, -21, right_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
98. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-24, 25] belongs to the function data. left=0/1, right=-2/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([2, 10, -2], [-24, 25], 0, Yes, -2, -2, Yes, -14, right_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 10, -2], [-24, 25], 0, Yes, -2, -2, Yes, -14, right_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
99. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 11, -1], not ascending-power order. The declared interval [lower,upper]=[-25, 26] belongs to the function data. left=1/1, right=-1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([3, 11, -1], [-25, 26], 1, Yes, 13, -1, Yes, -9, right_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 11, -1], [-25, 26], 1, Yes, 13, -1, Yes, -9, right_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
100. Compare Only Two Eligible Candidates, Not the Whole Problem
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 12, 0], not ascending-power order. The declared interval [lower,upper]=[-26, 27] belongs to the function data. left=2/1, right=0/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
View answer and explanation
([0, 12, 0], [-26, 27], 2, Yes, 24, 0, Yes, 0, right_lower, pairwise_only_no_global_optimum_claim)
Compute values even for ineligible points but compare only when both are feasible. A lower value among these two is not a global optimum certificate. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 12, 0], [-26, 27], 2, Yes, 24, 0, Yes, 0, right_lower, pairwise_only_no_global_optimum_claim).
Reviewed step-by-step derivation
Convex optimization problem · qf-mod25-convexopt-pairwise-feasible-comparison · candidate
Canonical question JSONL · English JSONL · Validation report