Question 25807: Construct the Dual from One Fixed Primal Convention and the Exact Transpose

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=[8, 9], c=[3, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, 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=[8, 9], c=[3, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, 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-construction-qualification]

derived = [[constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate]]

source roles = [[[1, 0], [0, 1]], [8, 9], [3, 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]

Reverse checks

• 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, [[1, 0], [0, 1]], [8, 9], [3, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)

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.