Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization

TL;DR

This paper presents a method to approximate nonnegative separable concave optimization problems with piecewise-linear functions, enabling the use of discrete optimization algorithms for efficient solutions with provable guarantees.

Contribution

It introduces a polynomial-size piecewise-linear approximation for concave problems over polyhedra, linking them to discrete optimization and enabling new algorithms and heuristics.

Findings

Achieved a polynomial bound on the number of pieces in the approximation.

Developed a new heuristic for concave multicommodity flow with 4.27% average error.

Designed a 1.4991+epsilon approximation algorithm for concave facility location.

Abstract

We study the problem of minimizing a nonnegative separable concave function over a compact feasible set. We approximate this problem to within a factor of 1+epsilon by a piecewise-linear minimization problem over the same feasible set. Our main result is that when the feasible set is a polyhedron, the number of resulting pieces is polynomial in the input size of the polyhedron and linear in 1/epsilon. For many practical concave cost problems, the resulting piecewise-linear cost problem can be formulated as a well-studied discrete optimization problem. As a result, a variety of polynomial-time exact algorithms, approximation algorithms, and polynomial-time heuristics for discrete optimization problems immediately yield fully polynomial-time approximation schemes, approximation algorithms, and polynomial-time heuristics for the corresponding concave cost problems. We illustrate our approach on two problems. For the concave cost multicommodity flow problem, we devise a new heuristic and study its performance using computational experiments. We are able to approximately solve significantly larger test instances than previously possible, and obtain solutions on average within 4.27% of optimality. For the concave cost facility location problem, we obtain a new 1.4991+epsilon approximation algorithm.