Overview — Optimisation

Pricing-error minimisation, quadratic utility, and constrained portfolio problems are the same mathematical object with different labels: find a critical point of a smooth objective, classify it with the Hessian, and — when a budget or risk budget binds — attach Lagrange multipliers (and, for inequalities, a KKT idea). This topic pulls unconstrained and constrained optimisation into one finance-facing unit without re-teaching the full mean–variance course; the portfolio constraint is the running example, not the curriculum destination.

By the end of this note you should be able to:

• locate critical points, apply the Hessian second-order test, and state a descent method (gradient or Newton);
• form a Lagrangian for equality constraints and interpret the multiplier as a shadow price;
• recognise convexity (Jensen / Hessian) so a local minimum is global, and state the KKT idea for simple inequalities.

1. Contents
2. The big picture