Second-Order Conditions

Optimization · Axiom Academy

LESSON Second-Order Conditions for Optimization Understanding why first-order conditions aren't enough and how the Hessian matrix determines optimality 1. The Problem with First-Order Conditions The first-order necessary condition for optimality is f(x*) = 0 . This identifies critical points, but critical points can be minima, maxima, or saddle points. Watch the animation to see why this is problematic: all three types of critical points satisfy the first-order condition! 2. Introducing the Hessian Matrix The Hessian matrix captures the second-order behavior of a function around a critical point. It's the matrix of all second partial derivatives. The Hessian describes the local curvature of the function. Watch how it relates to the shape of the function surface: 3. Sufficient Conditions for Local Minimum The key theorem for identifying local minima combines both first and second-order conditions: A matrix H is positive definite if v T Hv > 0 for all non-zero vectors v. This means the function curves upward in all directions. 4. Visualizing Minima, Maxima, and Saddle Points The classification of critical points depends on the definiteness of the Hessian: Positive definite H: Local minimum (bowl shape) Negative definite H: Local maximum (inverted bowl) Indefinite H: Saddle point (curves up in some directions, down in others) The definiteness of the Hessian can be checked through its eigenvalues. This provides a practical computational test for optimality.

This is the written version of the interactive lesson above. See the full Optimization course.