Question 29877: A Linear Constraint on a Complete Ordered Finite Point List

Original question and givens

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.

• G001 — 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.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-optimization-problem

Step-by-step derivation

source roles = [qf-app20-optimization-linear-constraint-audit, optimization-linear-constraint-audit]

derived = [[constraint, objective, direction, ALLrows, best, ALLarg_indices]]

Optimization ORIGINAL row0 constraint A times x := mul(2, 0)

Optimization ORIGINAL row0 constraint A times x = 0

Optimization ORIGINAL row0 constraint B times y := mul(1, 0)

Optimization ORIGINAL row0 constraint B times y = 0

Optimization ORIGINAL row0 constraint sum := sum(0, 0)

Optimization ORIGINAL row0 constraint sum = 0

Optimization ORIGINAL row0 objective u times x := mul(-1, 0)

Optimization ORIGINAL row0 objective u times x = 0

Optimization ORIGINAL row0 objective v times y := mul(2, 0)

Optimization ORIGINAL row0 objective v times y = 0

Optimization ORIGINAL row0 objective sum := sum(0, 0)

Optimization ORIGINAL row0 objective sum = 0

source roles = [0, [0, 0], [2, 1, 6], [-1, 2]]

derived = [0, 6, [0, 0, 0, Yes, 0]]

Optimization ORIGINAL row1 constraint A times x := mul(2, 1)

Optimization ORIGINAL row1 constraint A times x = 2

Optimization ORIGINAL row1 constraint B times y := mul(1, 1)

Optimization ORIGINAL row1 constraint B times y = 1

Optimization ORIGINAL row1 constraint sum := sum(2, 1)

Optimization ORIGINAL row1 constraint sum = 3

Optimization ORIGINAL row1 objective u times x := mul(-1, 1)

Optimization ORIGINAL row1 objective u times x = -1

Optimization ORIGINAL row1 objective v times y := mul(2, 1)

Optimization ORIGINAL row1 objective v times y = 2

Optimization ORIGINAL row1 objective sum := sum(-1, 2)

Optimization ORIGINAL row1 objective sum = 1

source roles = [1, [1, 1], [2, 1, 6], [-1, 2]]

derived = [3, 6, [1, 1, 1, Yes, 1]]

Optimization ORIGINAL row2 constraint A times x := mul(2, 2)

Optimization ORIGINAL row2 constraint A times x = 4

Optimization ORIGINAL row2 constraint B times y := mul(1, 2)

Optimization ORIGINAL row2 constraint B times y = 2

Optimization ORIGINAL row2 constraint sum := sum(4, 2)

Optimization ORIGINAL row2 constraint sum = 6

Optimization ORIGINAL row2 objective u times x := mul(-1, 2)

Optimization ORIGINAL row2 objective u times x = -2

Optimization ORIGINAL row2 objective v times y := mul(2, 2)

Optimization ORIGINAL row2 objective v times y = 4

Optimization ORIGINAL row2 objective sum := sum(-2, 4)

Optimization ORIGINAL row2 objective sum = 2

source roles = [2, [2, 2], [2, 1, 6], [-1, 2]]

derived = [6, 6, [2, 2, 2, Yes, 2]]

Optimization ORIGINAL row3 constraint A times x := mul(2, 3)

Optimization ORIGINAL row3 constraint A times x = 6

Optimization ORIGINAL row3 constraint B times y := mul(1, 1)

Optimization ORIGINAL row3 constraint B times y = 1

Optimization ORIGINAL row3 constraint sum := sum(6, 1)

Optimization ORIGINAL row3 constraint sum = 7

Optimization ORIGINAL row3 objective u times x := mul(-1, 3)

Optimization ORIGINAL row3 objective u times x = -3

Optimization ORIGINAL row3 objective v times y := mul(2, 1)

Optimization ORIGINAL row3 objective v times y = 2

Optimization ORIGINAL row3 objective sum := sum(-3, 2)

Optimization ORIGINAL row3 objective sum = -1

source roles = [3, [3, 1], [2, 1, 6], [-1, 2]]

derived = [7, 6, [3, 3, 1, No, -1]]

source roles = [linear finite candidate list, [0, 1, 2, -1], [0, 1, 2], min]

derived = [0, [0], [[0, 0, Yes, Yes], [1, 1, Yes, No], [2, 2, Yes, No]]]

source roles = [[[0, 0], [1, 1], [2, 2], [3, 1]], [2, 1, 6], [-1, 2], min]

derived = [[[0, 0, 0, Yes, 0], [1, 1, 1, Yes, 1], [2, 2, 2, Yes, 2], [3, 3, 1, No, -1]], 0, [0]]

Reverse checks

• S031 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S030 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S029 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S028 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 objective sum with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 objective v times y with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 objective u times x with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 constraint sum with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 constraint B times y with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 constraint A times x with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S021 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 objective sum with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 objective v times y with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 objective u times x with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 constraint sum with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 constraint B times y with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 constraint A times x with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S014 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 objective sum with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 objective v times y with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 objective u times x with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 constraint sum with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 constraint B times y with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 constraint A times x with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S007 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 objective sum with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 objective v times y with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 objective u times x with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 constraint sum with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 constraint B times y with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 constraint A times x with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.

Result

([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])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.