DC Decomposition of Nonconvex Polynomials with Algebraic Techniques

TL;DR

This paper introduces algebraic methods to decompose multivariate polynomials into differences of convex polynomials, enabling optimization via linear, second order cone, and semidefinite programming, with complexity results.

Contribution

It presents a novel algebraic approach to polynomial DC decomposition and analyzes the computational complexity of optimizing over all decompositions.

Findings

Decomposition reduces to linear, cone, and semidefinite programming.

Optimizing over all decompositions is NP-hard.

Methods improve speed of convex-concave procedures.

Abstract

We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) and find ones that speed up the convex-concave procedure (CCP). We prove, however, that optimizing over the entire set of dcds is NP-hard.