Question 27907: Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT

Original question and givens

Use mode=infeasible_point, complete model={"constraint_gradient":[1,7],"constraint_rhs":6,"point":[7,0]}. Give (mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary).

• G001 — Use mode=infeasible_point, complete model={"constraint_gradient":[1,7],"constraint_rhs":6,"point":[7,0]}. Give (mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary).

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-lagrange-multiplier-condition

Step-by-step derivation

source roles = [lagrange-boundary]

derived = [[mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary]]

Lagrange boundary ORIGINAL constraint value ORIGINAL product0 := mul(1, 7)

Lagrange boundary ORIGINAL constraint value ORIGINAL product0 = 7

Lagrange boundary ORIGINAL constraint value ORIGINAL running sum0 := sum(0, 7)

Lagrange boundary ORIGINAL constraint value ORIGINAL running sum0 = 7

Lagrange boundary ORIGINAL constraint value ORIGINAL product1 := mul(7, 0)

Lagrange boundary ORIGINAL constraint value ORIGINAL product1 = 0

Lagrange boundary ORIGINAL constraint value ORIGINAL running sum1 := sum(7, 0)

Lagrange boundary ORIGINAL constraint value ORIGINAL running sum1 = 7

source roles = [[1, 7], 6, [7, 0]]

derived = [7, No]

Reverse checks

• S006 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S005 · premise_and_result_audit — Recompute Lagrange boundary ORIGINAL constraint value ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Lagrange boundary ORIGINAL constraint value ORIGINAL product1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Lagrange boundary ORIGINAL constraint value ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Lagrange boundary ORIGINAL constraint value ORIGINAL product0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.

Result

(infeasible_point, [1, 7], 6, [7, 0], 7, No, stationarity_never_repairs_missing_feasibility)

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.