Convex Functions

Optimization · Axiom Academy

Understanding the fundamental properties and characterizations of convex functions in optimization 1. Definition of Convex Functions A function f is convex if its domain is a convex set and for all x, y in the domain and all [0,1] , the function value at any point on the line segment between x and y is at most the weighted average of the function values at the endpoints. The animation below illustrates this property: the chord connecting two points on the graph always lies above the function. An alternative definition uses the epigraph (the region above the graph). The epigraph of function f is the set of points that lie on or above its graph. Key Result: A function is convex if and only if its epigraph is a convex set. This provides a geometric way to verify convexity. For differentiable functions, convexity has an elegant characterization using gradients. A differentiable function is convex if and only if its graph lies above any tangent line. This says that the first-order Taylor approximation at any point x is a global underestimator of the function. The gradient provides a supporting hyperplane at every point. For twice-differentiable functions, we can use the Hessian matrix (matrix of second derivatives) to check convexity. A twice-differentiable function is convex if and only if its Hessian is positive semidefinite everywhere.

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