Convex Duality Made Difficult

TL;DR

This paper explores the categorical structure of convex optimization problems, aiming to develop a new theoretical framework for understanding and deriving results in convex analysis through category theory.

Contribution

It introduces a categorical approach to convex optimization problems, providing new methods to derive classical results like minimax theorems and properties of Legendre duality.

Findings

Re-derivation of a minimax-type theorem

Proof that (f*)* = f for convex functions

Development of a categorical framework for optimization

Abstract

The study of convex functions - in particular, of their optimization (really minimization) is one of the most important fields of applied mathematics. Convexity seems to be one of those incredibly well-chosen hypotheses which is just specific enough to admit a wealth of theorems, just general enough to produce a nontrivial theory (and a large amount of important examples). Convex optimization, possibly because it has an "analytical" rather than "algebraic" feel, has not been very thoroughly studied by applied category theorists. The one notable exception is [4], which studies the decomposition of optimization problems by categorical means. This paper takes a different approach, attempting to define a category with optimization problems as the objects, and to prove theorems about optimization by categorical means. As an illustration, we show how to use our methods to rederive some existing results: A minimax-type theorem, Theorem 5.5, and the fact that for convex functions, (f*)*=f (where f* is the Legendre dual), Proposition 6.6.