Question 25052: The Slack Equality Representation Preserves Feasibility in Both Directions

Original question and givens

Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[2, 1], [1, 2]], b=[5, 5], c=[3, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).

• G001 — Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[2, 1], [1, 2]], b=[5, 5], c=[3, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, 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-standard-form

Step-by-step derivation

source roles = [lp-standard-slack-conversion]

derived = [[x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate]]

source roles = [[[2, 1], [1, 2]], [5, 5], [3, 1]]

derived = [2, 2, maximize cTx subject Ax<=b,x>=0, Yes]

LP ORIGINAL constraint row0 product0 := mul(2, 1)

LP ORIGINAL constraint row0 product0 = 2

LP ORIGINAL constraint row0 cumulative0 := sum(0, 2)

LP ORIGINAL constraint row0 cumulative0 = 2

LP ORIGINAL constraint row0 product1 := mul(1, 1)

LP ORIGINAL constraint row0 product1 = 1

LP ORIGINAL constraint row0 cumulative1 := sum(2, 1)

LP ORIGINAL constraint row0 cumulative1 = 3

LP ORIGINAL constraint row1 product0 := mul(1, 1)

LP ORIGINAL constraint row1 product0 = 1

LP ORIGINAL constraint row1 cumulative0 := sum(0, 1)

LP ORIGINAL constraint row1 cumulative0 = 1

LP ORIGINAL constraint row1 product1 := mul(2, 1)

LP ORIGINAL constraint row1 product1 = 2

LP ORIGINAL constraint row1 cumulative1 := sum(1, 2)

LP ORIGINAL constraint row1 cumulative1 = 3

LP UNIQUE slack row0 := sub(5, 3)

LP UNIQUE slack row0 = 2

LP UNIQUE slack row1 := sub(5, 3)

LP UNIQUE slack row1 = 2

LP ORIGINAL objective product0 := mul(3, 1)

LP ORIGINAL objective product0 = 3

LP ORIGINAL objective cumulative0 := sum(0, 3)

LP ORIGINAL objective cumulative0 = 3

LP ORIGINAL objective product1 := mul(1, 1)

LP ORIGINAL objective product1 = 1

LP ORIGINAL objective cumulative1 := sum(3, 1)

LP ORIGINAL objective cumulative1 = 4

source roles = [[1, 1], [3, 3], [5, 5], [2, 2], [3, 1]]

derived = [Yes, Yes, Yes, Yes, Yes, 4]

Reverse checks

• 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 LP ORIGINAL objective cumulative1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute LP ORIGINAL objective product1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute LP ORIGINAL objective cumulative0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute LP ORIGINAL objective product0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute LP UNIQUE slack row1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute LP UNIQUE slack row0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute LP ORIGINAL constraint row1 cumulative1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute LP ORIGINAL constraint row1 product1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute LP ORIGINAL constraint row1 cumulative0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute LP ORIGINAL constraint row1 product0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute LP ORIGINAL constraint row0 cumulative1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute LP ORIGINAL constraint row0 product1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute LP ORIGINAL constraint row0 cumulative0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute LP ORIGINAL constraint 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

([1, 1], [3, 3], [5, 5], [2, 2], Yes, Yes, Yes, Yes, Yes, 4, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)

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.