Convex Optimization: Epigraphs

This lecture covers the concept of epigraphs, focusing on determining a class of functions whose epigraphs are halfspaces, polyhedra, and convex cones. It also explores the convexity of bivariate functions, analyzing functions like f(x1,x2) = 2x1x2 and f(x1,x2) = 1/(x1x2) on specific domains. Additionally, it delves into the convexity of log-sum-exp functions, crucial in logistic regression, by computing the Hessian in two and three dimensions and generalizing the results. The instructor provides rigorous arguments and hints for analysis.