Necessary Lagrange multiplier condition for equality constraints

Mathematical statement

Consider minimizing f(x) subject to g_1(x)=...=g_m(x)=0. The theorem premises include a feasible x*, differentiability of f and the constraints in a neighborhood, and linearly independent constraint-gradient rows at x*; the Jacobian has full row rank. The necessary local-extremum equation has a unique multiplier vector lambda with grad f(x*)=J_g(x*)^T*lambda, fixing this positive-sign convention. Exact vector families audit rank and the algebraic stationarity residual; they do not themselves establish feasibility, differentiability or a local extremum for functions not supplied there. A separate complete family solves squared distance to a center under one nonzero linear constraint. Stationarity is necessary, not sufficient. Rank deficiency withholds this regular theorem rather than implying no extremum. Equality multipliers are not inequality KKT conditions and do not imply a general global optimum or unique primal point.

Representative formula (exact TeX)

L(f,g,x)=l

Assumptions

• Constraint gradients are rows in declared order; m is at most the ambient dimension and exact rational row rank is m.
• A theorem application requires feasibility and regularity before stationarity; the vector audits use exact rationals.
• The completely solved single-constraint family has a nonzero normal and squared-distance objective.

Symbol scope

• L — Lagrange multiplier condition
• f — objective function
• g — equality-constraint map
• x — regular feasible point
• l — multiplier vector

Intuition, definition, and misconceptions

Feasibility, differentiability and full row rank come first.

Fix grad f=J_g^T*lambda to prevent sign and row/column convention drift.

Necessary stationarity is not sufficiency, KKT, general globality or primal uniqueness.

Solving for lambda does not prove a minimum, and dependent constraint gradients do not qualify the regular theorem.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Gradient vector — Use objective and constraint gradients.
• Optimization problem — Require a feasible local extremum.
• Linear systems in two variables and degenerate constraints — Retain the multiplier linear-system background.
• Matrix rank — Qualify full Jacobian row rank.
• Linear independence and bases — Specify constraint-gradient regularity.

Verification boundary

The general scope does not supply inequality KKT, second-order sufficient conditions or global-optimality certificates; the explicitly solved squared-distance instance remains a separate narrow problem.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add a tangent-space proof, KKT constraint qualifications and second-order tests.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.