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Convex Duality Made Difficult

Eigil Fjeldgren Rischel

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.

Convex Duality Made Difficult

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.
Paper Structure (6 sections, 22 theorems, 24 equations)

This paper contains 6 sections, 22 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Convex optimization
  3. Convex spaces
  4. The Category of Minmax problems
  5. Strong duality
  6. The Legendre Transform

Key Result

Proposition 3.4

A function between vector spaces is efr-002N if and only if it is a efr-0020.

Theorems & Definitions (51)

  • Definition 2.1: Standard form convex optimization problem
  • Definition 2.2: Lagrangian of an optimization problem
  • Definition 3.1: Convex Space
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Definition 3.5: Simplex
  • Definition 3.6: Topological convex space
  • Definition 4.1: Minmax problem
  • ...and 41 more