Question 2495: A Linear Objective Does Not by Itself Establish Feasibility

Original question and givens

Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 4], [1, 2]], objective [u,v]=[2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.

• G001 — Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 3], [2, 4], [1, 2]], objective [u,v]=[2, 2]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family.

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-linear-programming

Step-by-step derivation

explicit finite candidate grid and linear rows a·x+b·y≤c

keep candidate-grid optimization separate from continuous LP

[[0, 0], [1, 3], [2, 4], [1, 2]]

deduplicated canonical indexed points=[[0, 0, 0], [1, 1, 2], [2, 1, 3], [3, 2, 4]]

point0 objective first term := mul(2, 0)

point0 objective first term = 0

point0 objective second term := mul(2, 0)

point0 objective second term = 0

point0 objective := sum(0, 0)

point0 objective = 0

point1 objective first term := mul(2, 1)

point1 objective first term = 2

point1 objective second term := mul(2, 2)

point1 objective second term = 4

point1 objective := sum(2, 4)

point1 objective = 6

point2 objective first term := mul(2, 1)

point2 objective first term = 2

point2 objective second term := mul(2, 3)

point2 objective second term = 6

point2 objective := sum(2, 6)

point2 objective = 8

point3 objective first term := mul(2, 2)

point3 objective first term = 4

point3 objective second term := mul(2, 4)

point3 objective second term = 8

point3 objective := sum(4, 8)

point3 objective = 12

[[0, 0, 0, 0], [1, 1, 2, 6], [2, 1, 3, 8], [3, 2, 4, 12]]

objective values alone do not certify feasibility

finite candidates have been completely audited

no unlisted-point or general-LP optimality claim

Reverse checks

• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute point3 objective with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute point3 objective second term with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute point3 objective first term with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute point2 objective with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute point2 objective second term with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute point2 objective first term with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute point1 objective with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute point1 objective second term with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute point1 objective first term with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute point0 objective with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute point0 objective second term with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute point0 objective first term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

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

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.