Separable Concave Optimization Approximately Equals Piecewise Linear Optimization

Abstract

We show how to approximate a separable concave minimization problem over a general closed ground set by a single piecewise linear minimization problem. The approximation is to arbitrary 1+ε precision in optimal cost. For polyhedral ground sets in $\mathbb{R}^n_+$ and nondecreasing cost functions, the number of pieces is polynomial in the input size and proportional to 1/log(1+ε). For general polyhedra, the number of pieces is polynomial in the input size and the size of the zeroes of the concave objective components.

We illustrate our approach on the concave-cost uncapacitated multicommodity flow problem. By formulating the resulting piecewise linear approximation problem as a fixed charge, mixed integer model and using a dual ascent solution procedure, we solve randomly generated instances to within five to twenty percent of guaranteed optimality. The problem instances contain up to 50 nodes, 500 edges, 1,225 commodities, and 1,250,000 flow variables.