Algorithms for DC Programming via Polyhedral Approximations of Convex Functions

TL;DR

This paper introduces cutting-plane algorithms for approximating convex functions with polyhedral underestimators, enabling efficient solutions to DC programming problems with convergence guarantees and practical computational results.

Contribution

It develops new algorithms for generating polyhedral underestimators of convex functions, facilitating the solution of DC programming problems with proven correctness and convergence.

Findings

Algorithms effectively generate $ ext{ extepsilon}$-polyhedral underestimators.

The proposed methods converge to a global minimizer when $ ext{ extepsilon}$=0.

Computational tests demonstrate practical applicability.

Abstract

There is an existing exact algorithm that solves DC programming problems if one component of the DC function is polyhedral convex (Loehne, Wagner, 2017). Motivated by this, first, we consider two cutting-plane algorithms for generating an $\epsilon$-polyhedral underestimator of a convex function g. The algorithms start with a polyhedral underestimator of g and the epigraph of the current underestimator is intersected with either a single halfspace (Algorithm 1) or with possibly multiple halfspaces (Algorithm 2) in each iteration to obtain a better approximation. We prove the correctness and finiteness of both algorithms, establish the convergence rate of Algorithm 1, and show that after obtaining an $\epsilon$-polyhedral underestimator of the first component of a DC function, the algorithm from (Loehne, Wagner, 2017) can be applied to compute an $\epsilon$ solution of the DC programming problem without further computational effort. We then propose an algorithm (Algorithm 3) for solving DC programming problems by iteratively generating a (not necessarily $\epsilon$-) polyhedral underestimator of g. We prove that Algorithm 3 stops after finitely many iterations and it returns an $\epsilon$-solution to the DC programming problem. Moreover, the sequence $\{x_k\}_{k\geq 0} outputted by Algorithm 3 converges to a global minimizer of the DC problem when $\epsilon$ is set to zero. Computational results based on some test instances from the literature are provided.