Question 25867: Both Feasibility Checks Precede the Weak-Duality Inequality Chain

Original question and givens

Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[18, 19], c=[1, 2]. Candidate x=[18/2, 19/2], y=[2, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).

• G001 — Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[18, 19], c=[1, 2]. Candidate x=[18/2, 19/2], y=[2, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).

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-program-duality

Step-by-step derivation

source roles = [lpdual-weak-duality-chain]

derived = [[x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate]]

source roles = [[[1, 0], [0, 1]], [18, 19], [1, 2]]

derived = [2, 2, [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes]

LPdual ORIGINAL Ax row0 product0 := mul(1, 9)

LPdual ORIGINAL Ax row0 product0 = 9

LPdual ORIGINAL Ax row0 product1 := mul(0, 19/2)

LPdual ORIGINAL Ax row0 product1 = 0

LPdual ORIGINAL Ax row0 sum := sum(9, 0)

LPdual ORIGINAL Ax row0 sum = 9

LPdual ORIGINAL Ax row1 product0 := mul(0, 9)

LPdual ORIGINAL Ax row1 product0 = 0

LPdual ORIGINAL Ax row1 product1 := mul(1, 19/2)

LPdual ORIGINAL Ax row1 product1 = 19/2

LPdual ORIGINAL Ax row1 sum := sum(0, 19/2)

LPdual ORIGINAL Ax row1 sum = 19/2

LPdual ORIGINAL ATy row0 product0 := mul(1, 2)

LPdual ORIGINAL ATy row0 product0 = 2

LPdual ORIGINAL ATy row0 product1 := mul(0, 3)

LPdual ORIGINAL ATy row0 product1 = 0

LPdual ORIGINAL ATy row0 sum := sum(2, 0)

LPdual ORIGINAL ATy row0 sum = 2

LPdual ORIGINAL ATy row1 product0 := mul(0, 2)

LPdual ORIGINAL ATy row1 product0 = 0

LPdual ORIGINAL ATy row1 product1 := mul(1, 3)

LPdual ORIGINAL ATy row1 product1 = 3

LPdual ORIGINAL ATy row1 sum := sum(0, 3)

LPdual ORIGINAL ATy row1 sum = 3

LPdual ORIGINAL primal objective product0 := mul(1, 9)

LPdual ORIGINAL primal objective product0 = 9

LPdual ORIGINAL primal objective product1 := mul(2, 19/2)

LPdual ORIGINAL primal objective product1 = 19

LPdual ORIGINAL primal objective sum := sum(9, 19)

LPdual ORIGINAL primal objective sum = 28

LPdual ORIGINAL middle yTAx product0 := mul(2, 9)

LPdual ORIGINAL middle yTAx product0 = 18

LPdual ORIGINAL middle yTAx product1 := mul(3, 19/2)

LPdual ORIGINAL middle yTAx product1 = 57/2

LPdual ORIGINAL middle yTAx sum := sum(18, 57/2)

LPdual ORIGINAL middle yTAx sum = 93/2

LPdual ORIGINAL dual objective product0 := mul(18, 2)

LPdual ORIGINAL dual objective product0 = 36

LPdual ORIGINAL dual objective product1 := mul(19, 3)

LPdual ORIGINAL dual objective product1 = 57

LPdual ORIGINAL dual objective sum := sum(36, 57)

LPdual ORIGINAL dual objective sum = 93

source roles = [[9, 19/2], [2, 3], [9, 19/2], [2, 3], [18, 19], [1, 2]]

derived = [Yes, Yes, 28, 93/2, 93, [[0, 0, 2, 1, 9, 18], [0, 1, 2, 0, 19/2, 0], [1, 0, 3, 0, 9, 0], [1, 1, 3, 1, 19/2, 57/2]], 93/2, [[0, 9, 1, 9], [1, 19/2, 1, 19/2]], 37/2, [[0, 2, 9, 18], [1, 3, 19/2, 57/2]], 93/2, Yes, Yes, No]

Reverse checks

• 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 LPdual ORIGINAL dual objective sum with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute LPdual ORIGINAL dual objective product1 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute LPdual ORIGINAL dual objective product0 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute LPdual ORIGINAL middle yTAx sum with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute LPdual ORIGINAL middle yTAx product1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute LPdual ORIGINAL middle yTAx product0 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute LPdual ORIGINAL primal objective sum with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute LPdual ORIGINAL primal objective product1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute LPdual ORIGINAL primal objective product0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute LPdual ORIGINAL ATy row1 sum with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute LPdual ORIGINAL ATy row1 product1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute LPdual ORIGINAL ATy row1 product0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute LPdual ORIGINAL ATy row0 sum with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute LPdual ORIGINAL ATy row0 product1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute LPdual ORIGINAL ATy row0 product0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute LPdual ORIGINAL Ax row1 sum with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute LPdual ORIGINAL Ax row1 product1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute LPdual ORIGINAL Ax row1 product0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute LPdual ORIGINAL Ax row0 sum with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute LPdual ORIGINAL Ax row0 product1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute LPdual ORIGINAL Ax row0 product0 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

([9, 19/2], [2, 3], [9, 19/2], [2, 3], Yes, Yes, 28, 93/2, 93, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)

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.