Summary of constrained optimization

Unconstrained critical points.
- Solve $\nabla f=0$.
- Classify via the Hessian: positive definite (min), negative definite (max), indefinite (saddle).

Lagrange multipliers. For $f$ under $g=0$: $\nabla f=\lambda\nabla g$, plus the constraint.

Boundary. Parameterize or restrict $f$ to the boundary surface; combine with interior results.

Compact sets. If the set is compact, global extrema exist; check interior, boundary, and corners.

KKT for inequalities. For $g\le0$: Lagrange with nonnegativity conditions.

Examples. Circles, lines, ellipsoids, quadratics, multi-constraint problems.

Conclusion. Constrained optimization reduces to finding critical points of the Lagrangian and evaluating on boundaries.