Question 2464: All Tied Optimizers among the Explicit Feasible Candidates

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, 1], [2, 3], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]], objective=[1, 2], direction=min. 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. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).

• 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, 1], [2, 3], [1, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 9]], objective=[1, 2], direction=min. 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. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices).

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, 1], [2, 3], [1, 2]]

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

point0 objective first term := mul(1, 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

point0 row0 a·x := mul(-1, 0)

point0 row0 a·x = 0

point0 row0 b·y := mul(0, 0)

point0 row0 b·y = 0

point0 row0 LHS := sum(0, 0)

point0 row0 LHS = 0

row0: 0≤0

satisfied=Yes

point0 row1 a·x := mul(0, 0)

point0 row1 a·x = 0

point0 row1 b·y := mul(-1, 0)

point0 row1 b·y = 0

point0 row1 LHS := sum(0, 0)

point0 row1 LHS = 0

row1: 0≤0

satisfied=Yes

point0 row2 a·x := mul(1, 0)

point0 row2 a·x = 0

point0 row2 b·y := mul(1, 0)

point0 row2 b·y = 0

point0 row2 LHS := sum(0, 0)

point0 row2 LHS = 0

row2: 0≤9

satisfied=Yes

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

point1 objective first term = 1

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

point1 objective second term = 2

point1 objective := sum(1, 2)

point1 objective = 3

point1 row0 a·x := mul(-1, 1)

point1 row0 a·x = -1

point1 row0 b·y := mul(0, 1)

point1 row0 b·y = 0

point1 row0 LHS := sum(-1, 0)

point1 row0 LHS = -1

row0: -1≤0

satisfied=Yes

point1 row1 a·x := mul(0, 1)

point1 row1 a·x = 0

point1 row1 b·y := mul(-1, 1)

point1 row1 b·y = -1

point1 row1 LHS := sum(0, -1)

point1 row1 LHS = -1

row1: -1≤0

satisfied=Yes

point1 row2 a·x := mul(1, 1)

point1 row2 a·x = 1

point1 row2 b·y := mul(1, 1)

point1 row2 b·y = 1

point1 row2 LHS := sum(1, 1)

point1 row2 LHS = 2

row2: 2≤9

satisfied=Yes

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

point2 objective first term = 1

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

point2 objective second term = 4

point2 objective := sum(1, 4)

point2 objective = 5

point2 row0 a·x := mul(-1, 1)

point2 row0 a·x = -1

point2 row0 b·y := mul(0, 2)

point2 row0 b·y = 0

point2 row0 LHS := sum(-1, 0)

point2 row0 LHS = -1

row0: -1≤0

satisfied=Yes

point2 row1 a·x := mul(0, 1)

point2 row1 a·x = 0

point2 row1 b·y := mul(-1, 2)

point2 row1 b·y = -2

point2 row1 LHS := sum(0, -2)

point2 row1 LHS = -2

row1: -2≤0

satisfied=Yes

point2 row2 a·x := mul(1, 1)

point2 row2 a·x = 1

point2 row2 b·y := mul(1, 2)

point2 row2 b·y = 2

point2 row2 LHS := sum(1, 2)

point2 row2 LHS = 3

row2: 3≤9

satisfied=Yes

point3 objective first term := mul(1, 2)

point3 objective first term = 2

point3 objective second term := mul(2, 3)

point3 objective second term = 6

point3 objective := sum(2, 6)

point3 objective = 8

point3 row0 a·x := mul(-1, 2)

point3 row0 a·x = -2

point3 row0 b·y := mul(0, 3)

point3 row0 b·y = 0

point3 row0 LHS := sum(-2, 0)

point3 row0 LHS = -2

row0: -2≤0

satisfied=Yes

point3 row1 a·x := mul(0, 2)

point3 row1 a·x = 0

point3 row1 b·y := mul(-1, 3)

point3 row1 b·y = -3

point3 row1 LHS := sum(0, -3)

point3 row1 LHS = -3

row1: -3≤0

satisfied=Yes

point3 row2 a·x := mul(1, 2)

point3 row2 a·x = 2

point3 row2 b·y := mul(1, 3)

point3 row2 b·y = 3

point3 row2 LHS := sum(2, 3)

point3 row2 LHS = 5

row2: 5≤9

satisfied=Yes

direction=min; feasible objective rows=[[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 3], [2, 1, 2, [Yes, Yes, Yes], Yes, 5], [3, 2, 3, [Yes, Yes, Yes], Yes, 8]]

best among listed feasible points=0; all attaining indices=[0]

finite candidates have been completely audited

no unlisted-point or general-LP optimality claim

Reverse checks

• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Recompute point3 row2 LHS with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute point3 row2 b·y with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute point3 row2 a·x with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Recompute point3 row1 LHS with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute point3 row1 b·y with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute point3 row1 a·x with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Recompute point3 row0 LHS with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute point3 row0 b·y with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute point3 row0 a·x with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute point3 objective with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute point3 objective second term with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute point3 objective first term with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Recompute point2 row2 LHS with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute point2 row2 b·y with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute point2 row2 a·x with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Recompute point2 row1 LHS with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute point2 row1 b·y with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute point2 row1 a·x with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Recompute point2 row0 LHS with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute point2 row0 b·y with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute point2 row0 a·x with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute point2 objective with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute point2 objective second term with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute point2 objective first term with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Recompute point1 row2 LHS with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute point1 row2 b·y with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute point1 row2 a·x with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Recompute point1 row1 LHS with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute point1 row1 b·y with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute point1 row1 a·x with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Recompute point1 row0 LHS with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute point1 row0 b·y with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute point1 row0 a·x with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute point1 objective with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute point1 objective second term with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute point1 objective first term with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute point0 row2 LHS with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute point0 row2 b·y with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute point0 row2 a·x with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute point0 row1 LHS with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute point0 row1 b·y with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute point0 row1 a·x with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute point0 row0 LHS with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute point0 row0 b·y with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute point0 row0 a·x 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

([[-1, 0, 0], [0, -1, 0], [1, 1, 9]], [1, 2], min, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 1, 1, [Yes, Yes, Yes], Yes, 3], [2, 1, 2, [Yes, Yes, Yes], Yes, 5], [3, 2, 3, [Yes, Yes, Yes], Yes, 8]], 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.