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Complete Finite Feasibility and All Tied Optimizers

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batch-20261013-advance-20261020-r04-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. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-10, -8, 12, 0], feasible=[true, true, false, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([-10, -8, 12, 0], [Yes, Yes, No, Yes], [0, 1, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-10, -8, 12, 0], [Yes, Yes, No, Yes], [0, 1, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

2. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-9, -5, 11, 1], feasible=[true, false, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([-9, -5, 11, 1], [Yes, No, Yes, Yes], [0, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-9, -5, 11, 1], [Yes, No, Yes, Yes], [0, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

3. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-8, -2, 10, 2], feasible=[true, true, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([-8, -2, 10, 2], [Yes, Yes, Yes, Yes], [0, 1, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-8, -2, 10, 2], [Yes, Yes, Yes, Yes], [0, 1, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

4. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-7, 1, 9, 3], feasible=[true, false, false, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([-7, 1, 9, 3], [Yes, No, No, Yes], [0, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-7, 1, 9, 3], [Yes, No, No, Yes], [0, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

5. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-6, 4, 8, 4], feasible=[true, true, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([-6, 4, 8, 4], [Yes, Yes, Yes, Yes], [0, 1, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-6, 4, 8, 4], [Yes, Yes, Yes, Yes], [0, 1, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

6. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-5, 7, 7, 0], feasible=[true, false, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([-5, 7, 7, 0], [Yes, No, Yes, Yes], [0, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-5, 7, 7, 0], [Yes, No, Yes, Yes], [0, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

7. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-4, -7, 6, 1], feasible=[true, true, false, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([-4, -7, 6, 1], [Yes, Yes, No, Yes], [0, 1, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-4, -7, 6, 1], [Yes, Yes, No, Yes], [0, 1, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

8. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-3, -4, 5, 2], feasible=[true, false, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([-3, -4, 5, 2], [Yes, No, Yes, Yes], [0, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -4, 5, 2], [Yes, No, Yes, Yes], [0, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

9. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-2, -1, 4, 3], feasible=[true, true, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([-2, -1, 4, 3], [Yes, Yes, Yes, Yes], [0, 1, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, 4, 3], [Yes, Yes, Yes, Yes], [0, 1, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

10. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-1, 2, 3, 4], feasible=[true, false, false, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([-1, 2, 3, 4], [Yes, No, No, Yes], [0, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 3, 4], [Yes, No, No, Yes], [0, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

11. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[0, 5, 2, 0], feasible=[true, true, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([0, 5, 2, 0], [Yes, Yes, Yes, Yes], [0, 1, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 5, 2, 0], [Yes, Yes, Yes, Yes], [0, 1, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

12. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[1, 8, 1, 1], feasible=[true, false, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([1, 8, 1, 1], [Yes, No, Yes, Yes], [0, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 8, 1, 1], [Yes, No, Yes, Yes], [0, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

13. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[2, -6, 0, 2], feasible=[true, true, false, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([2, -6, 0, 2], [Yes, Yes, No, Yes], [0, 1, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -6, 0, 2], [Yes, Yes, No, Yes], [0, 1, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

14. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[3, -3, -1, 3], feasible=[true, false, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([3, -3, -1, 3], [Yes, No, Yes, Yes], [0, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -3, -1, 3], [Yes, No, Yes, Yes], [0, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

15. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[4, 0, -2, 4], feasible=[true, true, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([4, 0, -2, 4], [Yes, Yes, Yes, Yes], [0, 1, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4, 0, -2, 4], [Yes, Yes, Yes, Yes], [0, 1, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

16. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[5, 3, -3, 0], feasible=[true, false, false, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([5, 3, -3, 0], [Yes, No, No, Yes], [0, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([5, 3, -3, 0], [Yes, No, No, Yes], [0, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

17. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[6, 6, -4, 1], feasible=[true, true, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([6, 6, -4, 1], [Yes, Yes, Yes, Yes], [0, 1, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([6, 6, -4, 1], [Yes, Yes, Yes, Yes], [0, 1, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

18. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[7, -8, -5, 2], feasible=[true, false, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([7, -8, -5, 2], [Yes, No, Yes, Yes], [0, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([7, -8, -5, 2], [Yes, No, Yes, Yes], [0, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

19. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[8, -5, -6, 3], feasible=[true, true, false, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([8, -5, -6, 3], [Yes, Yes, No, Yes], [0, 1, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([8, -5, -6, 3], [Yes, Yes, No, Yes], [0, 1, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

20. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[9, -2, -7, 4], feasible=[true, false, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([9, -2, -7, 4], [Yes, No, Yes, Yes], [0, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([9, -2, -7, 4], [Yes, No, Yes, Yes], [0, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

21. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[10, 1, -8, 0], feasible=[true, true, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([10, 1, -8, 0], [Yes, Yes, Yes, Yes], [0, 1, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([10, 1, -8, 0], [Yes, Yes, Yes, Yes], [0, 1, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

22. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[11, 4, -9, 1], feasible=[true, false, false, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([11, 4, -9, 1], [Yes, No, No, Yes], [0, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([11, 4, -9, 1], [Yes, No, No, Yes], [0, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

23. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[12, 7, -10, 2], feasible=[true, true, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([12, 7, -10, 2], [Yes, Yes, Yes, Yes], [0, 1, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([12, 7, -10, 2], [Yes, Yes, Yes, Yes], [0, 1, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

24. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[13, -7, -11, 3], feasible=[true, false, true, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([13, -7, -11, 3], [Yes, No, Yes, Yes], [0, 2, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([13, -7, -11, 3], [Yes, No, Yes, Yes], [0, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

25. Feasible Original Indices before Objective Comparison

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[14, -4, -12, 4], feasible=[true, true, false, true]. Give (all values, feasible mask, every feasible original index).

View answer and explanation

([14, -4, -12, 4], [Yes, Yes, No, Yes], [0, 1, 3])

Filter by declared eligibility without selecting an infeasible point that has a better objective value. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([14, -4, -12, 4], [Yes, Yes, No, Yes], [0, 1, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-feasible-filter · candidate

26. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-10, -8, 12, 0], feasible=[true, true, false, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([-10, -8, 12, 0], [Yes, Yes, No, Yes], min, -10, [0])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-10, -8, 12, 0], [Yes, Yes, No, Yes], min, -10, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

27. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-9, -5, 11, 1], feasible=[true, false, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([-9, -5, 11, 1], [Yes, No, Yes, Yes], min, -9, [0])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-9, -5, 11, 1], [Yes, No, Yes, Yes], min, -9, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

28. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-8, -2, 10, 2], feasible=[true, true, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([-8, -2, 10, 2], [Yes, Yes, Yes, Yes], min, -8, [0])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-8, -2, 10, 2], [Yes, Yes, Yes, Yes], min, -8, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

29. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-7, 1, 9, 3], feasible=[true, false, false, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([-7, 1, 9, 3], [Yes, No, No, Yes], min, -7, [0])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-7, 1, 9, 3], [Yes, No, No, Yes], min, -7, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

30. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-6, 4, 8, 4], feasible=[true, true, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([-6, 4, 8, 4], [Yes, Yes, Yes, Yes], min, -6, [0])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-6, 4, 8, 4], [Yes, Yes, Yes, Yes], min, -6, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

31. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-5, 7, 7, 0], feasible=[true, false, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([-5, 7, 7, 0], [Yes, No, Yes, Yes], min, -5, [0])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-5, 7, 7, 0], [Yes, No, Yes, Yes], min, -5, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

32. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-4, -7, 6, 1], feasible=[true, true, false, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([-4, -7, 6, 1], [Yes, Yes, No, Yes], min, -7, [1])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-4, -7, 6, 1], [Yes, Yes, No, Yes], min, -7, [1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

33. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-3, -4, 5, 2], feasible=[true, false, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([-3, -4, 5, 2], [Yes, No, Yes, Yes], min, -3, [0])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -4, 5, 2], [Yes, No, Yes, Yes], min, -3, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

34. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-2, -1, 4, 3], feasible=[true, true, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([-2, -1, 4, 3], [Yes, Yes, Yes, Yes], min, -2, [0])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, 4, 3], [Yes, Yes, Yes, Yes], min, -2, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

35. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-1, 2, 3, 4], feasible=[true, false, false, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([-1, 2, 3, 4], [Yes, No, No, Yes], min, -1, [0])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 3, 4], [Yes, No, No, Yes], min, -1, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

36. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[0, 5, 2, 0], feasible=[true, true, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([0, 5, 2, 0], [Yes, Yes, Yes, Yes], min, 0, [0, 3])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 5, 2, 0], [Yes, Yes, Yes, Yes], min, 0, [0, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

37. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[1, 8, 1, 1], feasible=[true, false, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([1, 8, 1, 1], [Yes, No, Yes, Yes], min, 1, [0, 2, 3])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 8, 1, 1], [Yes, No, Yes, Yes], min, 1, [0, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

38. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[2, -6, 0, 2], feasible=[true, true, false, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([2, -6, 0, 2], [Yes, Yes, No, Yes], min, -6, [1])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -6, 0, 2], [Yes, Yes, No, Yes], min, -6, [1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

39. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[3, -3, -1, 3], feasible=[true, false, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([3, -3, -1, 3], [Yes, No, Yes, Yes], min, -1, [2])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -3, -1, 3], [Yes, No, Yes, Yes], min, -1, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

40. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[4, 0, -2, 4], feasible=[true, true, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([4, 0, -2, 4], [Yes, Yes, Yes, Yes], min, -2, [2])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4, 0, -2, 4], [Yes, Yes, Yes, Yes], min, -2, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

41. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[5, 3, -3, 0], feasible=[true, false, false, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([5, 3, -3, 0], [Yes, No, No, Yes], min, 0, [3])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([5, 3, -3, 0], [Yes, No, No, Yes], min, 0, [3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

42. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[6, 6, -4, 1], feasible=[true, true, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([6, 6, -4, 1], [Yes, Yes, Yes, Yes], min, -4, [2])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([6, 6, -4, 1], [Yes, Yes, Yes, Yes], min, -4, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

43. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[7, -8, -5, 2], feasible=[true, false, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([7, -8, -5, 2], [Yes, No, Yes, Yes], min, -5, [2])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([7, -8, -5, 2], [Yes, No, Yes, Yes], min, -5, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

44. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[8, -5, -6, 3], feasible=[true, true, false, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([8, -5, -6, 3], [Yes, Yes, No, Yes], min, -5, [1])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([8, -5, -6, 3], [Yes, Yes, No, Yes], min, -5, [1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

45. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[9, -2, -7, 4], feasible=[true, false, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([9, -2, -7, 4], [Yes, No, Yes, Yes], min, -7, [2])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([9, -2, -7, 4], [Yes, No, Yes, Yes], min, -7, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

46. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[10, 1, -8, 0], feasible=[true, true, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([10, 1, -8, 0], [Yes, Yes, Yes, Yes], min, -8, [2])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([10, 1, -8, 0], [Yes, Yes, Yes, Yes], min, -8, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

47. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[11, 4, -9, 1], feasible=[true, false, false, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([11, 4, -9, 1], [Yes, No, No, Yes], min, 1, [3])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([11, 4, -9, 1], [Yes, No, No, Yes], min, 1, [3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

48. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[12, 7, -10, 2], feasible=[true, true, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([12, 7, -10, 2], [Yes, Yes, Yes, Yes], min, -10, [2])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([12, 7, -10, 2], [Yes, Yes, Yes, Yes], min, -10, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

49. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[13, -7, -11, 3], feasible=[true, false, true, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([13, -7, -11, 3], [Yes, No, Yes, Yes], min, -11, [2])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([13, -7, -11, 3], [Yes, No, Yes, Yes], min, -11, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

50. Finite Minimum Value and Every Original Minimizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[14, -4, -12, 4], feasible=[true, true, false, true]. Give (values, mask, min, minimum feasible value, all minimizing original indices).

View answer and explanation

([14, -4, -12, 4], [Yes, Yes, No, Yes], min, -4, [1])

Separate the optimal numerical value from the complete tied argmin set. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([14, -4, -12, 4], [Yes, Yes, No, Yes], min, -4, [1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmin · candidate

51. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-10, -8, 12, 0], feasible=[true, true, false, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([-10, -8, 12, 0], [Yes, Yes, No, Yes], max, 0, [3])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-10, -8, 12, 0], [Yes, Yes, No, Yes], max, 0, [3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

52. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-9, -5, 11, 1], feasible=[true, false, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([-9, -5, 11, 1], [Yes, No, Yes, Yes], max, 11, [2])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-9, -5, 11, 1], [Yes, No, Yes, Yes], max, 11, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

53. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-8, -2, 10, 2], feasible=[true, true, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([-8, -2, 10, 2], [Yes, Yes, Yes, Yes], max, 10, [2])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-8, -2, 10, 2], [Yes, Yes, Yes, Yes], max, 10, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

54. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-7, 1, 9, 3], feasible=[true, false, false, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([-7, 1, 9, 3], [Yes, No, No, Yes], max, 3, [3])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-7, 1, 9, 3], [Yes, No, No, Yes], max, 3, [3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

55. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-6, 4, 8, 4], feasible=[true, true, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([-6, 4, 8, 4], [Yes, Yes, Yes, Yes], max, 8, [2])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-6, 4, 8, 4], [Yes, Yes, Yes, Yes], max, 8, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

56. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-5, 7, 7, 0], feasible=[true, false, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([-5, 7, 7, 0], [Yes, No, Yes, Yes], max, 7, [2])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-5, 7, 7, 0], [Yes, No, Yes, Yes], max, 7, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

57. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-4, -7, 6, 1], feasible=[true, true, false, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([-4, -7, 6, 1], [Yes, Yes, No, Yes], max, 1, [3])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-4, -7, 6, 1], [Yes, Yes, No, Yes], max, 1, [3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

58. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-3, -4, 5, 2], feasible=[true, false, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([-3, -4, 5, 2], [Yes, No, Yes, Yes], max, 5, [2])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -4, 5, 2], [Yes, No, Yes, Yes], max, 5, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

59. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-2, -1, 4, 3], feasible=[true, true, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([-2, -1, 4, 3], [Yes, Yes, Yes, Yes], max, 4, [2])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, 4, 3], [Yes, Yes, Yes, Yes], max, 4, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

60. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[-1, 2, 3, 4], feasible=[true, false, false, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([-1, 2, 3, 4], [Yes, No, No, Yes], max, 4, [3])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 3, 4], [Yes, No, No, Yes], max, 4, [3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

61. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[0, 5, 2, 0], feasible=[true, true, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([0, 5, 2, 0], [Yes, Yes, Yes, Yes], max, 5, [1])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 5, 2, 0], [Yes, Yes, Yes, Yes], max, 5, [1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

62. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[1, 8, 1, 1], feasible=[true, false, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([1, 8, 1, 1], [Yes, No, Yes, Yes], max, 1, [0, 2, 3])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 8, 1, 1], [Yes, No, Yes, Yes], max, 1, [0, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

63. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[2, -6, 0, 2], feasible=[true, true, false, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([2, -6, 0, 2], [Yes, Yes, No, Yes], max, 2, [0, 3])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -6, 0, 2], [Yes, Yes, No, Yes], max, 2, [0, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

64. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[3, -3, -1, 3], feasible=[true, false, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([3, -3, -1, 3], [Yes, No, Yes, Yes], max, 3, [0, 3])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -3, -1, 3], [Yes, No, Yes, Yes], max, 3, [0, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

65. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[4, 0, -2, 4], feasible=[true, true, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([4, 0, -2, 4], [Yes, Yes, Yes, Yes], max, 4, [0, 3])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4, 0, -2, 4], [Yes, Yes, Yes, Yes], max, 4, [0, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

66. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[5, 3, -3, 0], feasible=[true, false, false, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([5, 3, -3, 0], [Yes, No, No, Yes], max, 5, [0])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([5, 3, -3, 0], [Yes, No, No, Yes], max, 5, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

67. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[6, 6, -4, 1], feasible=[true, true, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([6, 6, -4, 1], [Yes, Yes, Yes, Yes], max, 6, [0, 1])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([6, 6, -4, 1], [Yes, Yes, Yes, Yes], max, 6, [0, 1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

68. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[7, -8, -5, 2], feasible=[true, false, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([7, -8, -5, 2], [Yes, No, Yes, Yes], max, 7, [0])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([7, -8, -5, 2], [Yes, No, Yes, Yes], max, 7, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

69. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[8, -5, -6, 3], feasible=[true, true, false, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([8, -5, -6, 3], [Yes, Yes, No, Yes], max, 8, [0])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([8, -5, -6, 3], [Yes, Yes, No, Yes], max, 8, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

70. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[9, -2, -7, 4], feasible=[true, false, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([9, -2, -7, 4], [Yes, No, Yes, Yes], max, 9, [0])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([9, -2, -7, 4], [Yes, No, Yes, Yes], max, 9, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

71. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[10, 1, -8, 0], feasible=[true, true, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([10, 1, -8, 0], [Yes, Yes, Yes, Yes], max, 10, [0])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([10, 1, -8, 0], [Yes, Yes, Yes, Yes], max, 10, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

72. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[11, 4, -9, 1], feasible=[true, false, false, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([11, 4, -9, 1], [Yes, No, No, Yes], max, 11, [0])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([11, 4, -9, 1], [Yes, No, No, Yes], max, 11, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

73. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[12, 7, -10, 2], feasible=[true, true, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([12, 7, -10, 2], [Yes, Yes, Yes, Yes], max, 12, [0])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([12, 7, -10, 2], [Yes, Yes, Yes, Yes], max, 12, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

74. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[13, -7, -11, 3], feasible=[true, false, true, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([13, -7, -11, 3], [Yes, No, Yes, Yes], max, 13, [0])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([13, -7, -11, 3], [Yes, No, Yes, Yes], max, 13, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

75. Finite Maximum Value and Every Original Maximizer Index

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. values=[14, -4, -12, 4], feasible=[true, true, false, true]. Give (values, mask, max, maximum feasible value, all maximizing original indices).

View answer and explanation

([14, -4, -12, 4], [Yes, Yes, No, Yes], max, 14, [0])

Compare only feasible candidates and retain all tied indices, not a claim of uniqueness. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([14, -4, -12, 4], [Yes, Yes, No, Yes], max, 14, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-argmax · candidate

76. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 0], [2, 0], [3, 0]], constraint [A,B,c]=[1, 1, 5] means A*x+B*y≤c, objective [u,v]=[-2, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([1, 1, 5], [-2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 0, Yes, -2], [2, 2, 0, Yes, -4], [3, 3, 0, Yes, -6]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, 5], [-2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 0, Yes, -2], [2, 2, 0, Yes, -4], [3, 3, 0, Yes, -6]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

77. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 1], [2, 2], [3, 1]], constraint [A,B,c]=[2, 1, 6] means A*x+B*y≤c, objective [u,v]=[-1, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([2, 1, 6], [-1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 2, Yes, 2], [3, 3, 1, No, -1]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 6], [-1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 2, Yes, 2], [3, 3, 1, No, -1]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

78. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 2], [2, 4], [3, 2]], constraint [A,B,c]=[3, 1, 7] means A*x+B*y≤c, objective [u,v]=[0, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([3, 1, 7], [0, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 4, No, 8], [3, 3, 2, No, 4]], 4, [1])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 7], [0, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 4, No, 8], [3, 3, 2, No, 4]], 4, [1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

79. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 3], [2, 6], [3, 3]], constraint [A,B,c]=[1, 1, 8] means A*x+B*y≤c, objective [u,v]=[1, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([1, 1, 8], [1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 3, Yes, 7], [2, 2, 6, Yes, 14], [3, 3, 3, Yes, 9]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, 8], [1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 3, Yes, 7], [2, 2, 6, Yes, 14], [3, 3, 3, Yes, 9]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

80. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 4], [2, 1], [3, 0]], constraint [A,B,c]=[2, 1, 9] means A*x+B*y≤c, objective [u,v]=[2, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([2, 1, 9], [2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 4, Yes, 10], [2, 2, 1, Yes, 6], [3, 3, 0, Yes, 6]], 10, [1])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 9], [2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 4, Yes, 10], [2, 2, 1, Yes, 6], [3, 3, 0, Yes, 6]], 10, [1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

81. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 0], [2, 3], [3, 1]], constraint [A,B,c]=[3, 1, 10] means A*x+B*y≤c, objective [u,v]=[-2, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([3, 1, 10], [-2, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 0, Yes, -2], [2, 2, 3, Yes, 2], [3, 3, 1, Yes, -4]], -4, [3])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 10], [-2, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 0, Yes, -2], [2, 2, 3, Yes, 2], [3, 3, 1, Yes, -4]], -4, [3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

82. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 1], [2, 5], [3, 2]], constraint [A,B,c]=[1, 1, 11] means A*x+B*y≤c, objective [u,v]=[-1, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([1, 1, 11], [-1, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 5, Yes, 8], [3, 3, 2, Yes, 1]], 8, [2])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, 11], [-1, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 5, Yes, 8], [3, 3, 2, Yes, 1]], 8, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

83. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 2], [2, 0], [3, 3]], constraint [A,B,c]=[2, 1, 12] means A*x+B*y≤c, objective [u,v]=[0, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([2, 1, 12], [0, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 0, Yes, 0], [3, 3, 3, Yes, 6]], 0, [0, 2])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 12], [0, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 0, Yes, 0], [3, 3, 3, Yes, 6]], 0, [0, 2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

84. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 3], [2, 2], [3, 0]], constraint [A,B,c]=[3, 1, 5] means A*x+B*y≤c, objective [u,v]=[1, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([3, 1, 5], [1, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 3, No, 7], [2, 2, 2, No, 6], [3, 3, 0, No, 3]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 5], [1, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 3, No, 7], [2, 2, 2, No, 6], [3, 3, 0, No, 3]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

85. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 4], [2, 4], [3, 1]], constraint [A,B,c]=[1, 1, 6] means A*x+B*y≤c, objective [u,v]=[2, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([1, 1, 6], [2, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 4, Yes, 10], [2, 2, 4, Yes, 12], [3, 3, 1, Yes, 8]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, 6], [2, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 4, Yes, 10], [2, 2, 4, Yes, 12], [3, 3, 1, Yes, 8]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

86. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 0], [2, 6], [3, 2]], constraint [A,B,c]=[2, 1, 7] means A*x+B*y≤c, objective [u,v]=[-2, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([2, 1, 7], [-2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 0, Yes, -2], [2, 2, 6, No, 8], [3, 3, 2, No, -2]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 7], [-2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 0, Yes, -2], [2, 2, 6, No, 8], [3, 3, 2, No, -2]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

87. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 1], [2, 1], [3, 3]], constraint [A,B,c]=[3, 1, 8] means A*x+B*y≤c, objective [u,v]=[-1, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([3, 1, 8], [-1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 1, Yes, 0], [3, 3, 3, No, 3]], 0, [0, 2])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 8], [-1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 1, Yes, 0], [3, 3, 3, No, 3]], 0, [0, 2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

88. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 2], [2, 3], [3, 0]], constraint [A,B,c]=[1, 1, 9] means A*x+B*y≤c, objective [u,v]=[0, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([1, 1, 9], [0, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 3, Yes, 6], [3, 3, 0, Yes, 0]], 6, [2])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, 9], [0, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 3, Yes, 6], [3, 3, 0, Yes, 0]], 6, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

89. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 3], [2, 5], [3, 1]], constraint [A,B,c]=[2, 1, 10] means A*x+B*y≤c, objective [u,v]=[1, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([2, 1, 10], [1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 3, Yes, 7], [2, 2, 5, Yes, 12], [3, 3, 1, Yes, 5]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 10], [1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 3, Yes, 7], [2, 2, 5, Yes, 12], [3, 3, 1, Yes, 5]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

90. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 4], [2, 0], [3, 2]], constraint [A,B,c]=[3, 1, 11] means A*x+B*y≤c, objective [u,v]=[2, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([3, 1, 11], [2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 4, Yes, 10], [2, 2, 0, Yes, 4], [3, 3, 2, Yes, 10]], 10, [1, 3])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 11], [2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 4, Yes, 10], [2, 2, 0, Yes, 4], [3, 3, 2, Yes, 10]], 10, [1, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

91. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 0], [2, 2], [3, 3]], constraint [A,B,c]=[1, 1, 12] means A*x+B*y≤c, objective [u,v]=[-2, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([1, 1, 12], [-2, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 0, Yes, -2], [2, 2, 2, Yes, 0], [3, 3, 3, Yes, 0]], -2, [1])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, 12], [-2, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 0, Yes, -2], [2, 2, 2, Yes, 0], [3, 3, 3, Yes, 0]], -2, [1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

92. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 1], [2, 4], [3, 0]], constraint [A,B,c]=[2, 1, 5] means A*x+B*y≤c, objective [u,v]=[-1, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([2, 1, 5], [-1, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 4, No, 6], [3, 3, 0, No, -3]], 1, [1])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 5], [-1, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 4, No, 6], [3, 3, 0, No, -3]], 1, [1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

93. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 2], [2, 6], [3, 1]], constraint [A,B,c]=[3, 1, 6] means A*x+B*y≤c, objective [u,v]=[0, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([3, 1, 6], [0, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 6, No, 12], [3, 3, 1, No, 2]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 6], [0, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 6, No, 12], [3, 3, 1, No, 2]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

94. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 3], [2, 1], [3, 2]], constraint [A,B,c]=[1, 1, 7] means A*x+B*y≤c, objective [u,v]=[1, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([1, 1, 7], [1, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 3, Yes, 7], [2, 2, 1, Yes, 4], [3, 3, 2, Yes, 7]], 7, [1, 3])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, 7], [1, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 3, Yes, 7], [2, 2, 1, Yes, 4], [3, 3, 2, Yes, 7]], 7, [1, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

95. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 4], [2, 3], [3, 3]], constraint [A,B,c]=[2, 1, 8] means A*x+B*y≤c, objective [u,v]=[2, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([2, 1, 8], [2, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 4, Yes, 10], [2, 2, 3, Yes, 10], [3, 3, 3, No, 12]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 8], [2, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 4, Yes, 10], [2, 2, 3, Yes, 10], [3, 3, 3, No, 12]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

96. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 0], [2, 5], [3, 0]], constraint [A,B,c]=[3, 1, 9] means A*x+B*y≤c, objective [u,v]=[-2, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([3, 1, 9], [-2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 0, Yes, -2], [2, 2, 5, No, 6], [3, 3, 0, Yes, -6]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 9], [-2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 0, Yes, -2], [2, 2, 5, No, 6], [3, 3, 0, Yes, -6]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

97. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 1], [2, 0], [3, 1]], constraint [A,B,c]=[1, 1, 10] means A*x+B*y≤c, objective [u,v]=[-1, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([1, 1, 10], [-1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 0, Yes, -2], [3, 3, 1, Yes, -1]], -2, [2])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, 10], [-1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 0, Yes, -2], [3, 3, 1, Yes, -1]], -2, [2]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

98. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 2], [2, 2], [3, 2]], constraint [A,B,c]=[2, 1, 11] means A*x+B*y≤c, objective [u,v]=[0, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([2, 1, 11], [0, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 2, Yes, 4], [3, 3, 2, Yes, 4]], 4, [1, 2, 3])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 11], [0, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 2, Yes, 4], [3, 3, 2, Yes, 4]], 4, [1, 2, 3]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

99. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 3], [2, 4], [3, 3]], constraint [A,B,c]=[3, 1, 12] means A*x+B*y≤c, objective [u,v]=[1, 2] means u*x+v*y, direction=min. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([3, 1, 12], [1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 3, Yes, 7], [2, 2, 4, Yes, 10], [3, 3, 3, Yes, 9]], 0, [0])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 12], [1, 2], min, [[0, 0, 0, Yes, 0], [1, 1, 3, Yes, 7], [2, 2, 4, Yes, 10], [3, 3, 3, Yes, 9]], 0, [0]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

100. A Linear Constraint on a Complete Ordered Finite Point List

Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 4], [2, 6], [3, 0]], constraint [A,B,c]=[1, 1, 5] means A*x+B*y≤c, objective [u,v]=[2, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points.

View answer and explanation

([1, 1, 5], [2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 4, Yes, 10], [2, 2, 6, No, 16], [3, 3, 0, Yes, 6]], 10, [1])

Evaluate every row but optimize only over feasible rows; the result concerns this explicit list, not the whole continuous half-plane. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, 5], [2, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 4, Yes, 10], [2, 2, 6, No, 16], [3, 3, 0, Yes, 6]], 10, [1]).

Reviewed step-by-step derivation

Optimization problem · qf-app20-optimization-linear-constraint-audit · candidate

Canonical question JSONL · English JSONL · Validation report