Copositive Duality for Discrete Markets and Games

TL;DR

This paper leverages copositive duality to address nonconvex discrete optimization problems, applying it to power system pricing and game theory, and introduces a cutting plane algorithm for exact solutions.

Contribution

It introduces a novel approach using copositive duality for discrete markets and games, including new formulations and solution methods for these problems.

Findings

Successfully reformulated power system unit commitment as a copositive program

Developed a dual copositive pricing mechanism for power markets

Designed a cutting plane algorithm for solving copositive programs exactly

Abstract

Optimization problems with discrete decisions are nonconvex and thus lack strong duality, which limits the usefulness of tools such as shadow prices and the KKT conditions. It was shown in Burer(2009) that mixed-binary quadratic programs can be written as completely positive programs, which are convex. Completely positive reformulations of discrete optimization problems therefore have strong duality if a constraint qualification is satisfied. We apply this perspective in two ways. First, we write unit commitment in power systems as a completely positive program, and use the dual copositive program to design a new pricing mechanism. Second, we reformulate integer programming games in terms of completely positive programming, and use the KKT conditions to solve for pure strategy Nash equilibria. To facilitate implementation, we also design a cutting plane algorithm for solving copositive programs exactly.