Fenchel Conjugation: Basics and Applications

This lecture covers the concept of Fenchel conjugation, which allows representing convex functions in max-form. It explores the properties and examples of Fenchel conjugates, including the conjugate of the conjugate. The lecture also discusses the application of Fenchel conjugation in solving nonsmooth optimization problems, such as composite minimization. Examples like l₁-norm and l₂-norm-squared functions are presented to illustrate the concept. Additionally, it delves into the connection between minimax optimization and Fenchel conjugation, showcasing how the conjugation reveals the underlying minimax problem. Various templates and formulations are discussed, providing insights into the versatility of Fenchel conjugation in mathematical optimization.