Absolute value linear programming

TL;DR

This paper explores the properties, complexity, and geometric structure of linear programming problems involving absolute values, revealing their nonconvexity, NP-hardness, and how they relate to nonconvex polyhedral sets.

Contribution

It provides fundamental insights into the topology, convexity, and complexity of absolute value linear programming problems, including conditions for solution optimality and the representation of nonconvex sets.

Findings

Feasible sets are nonconvex polyhedral sets.

Many basic questions are NP-hard to solve.

Absolute value constraints can describe all nonconvex polyhedral sets.

Abstract

We deal with linear programming problems involving absolute values in their formulations, so that they are no more expressible as standard linear programs. The presence of absolute values causes the problems to be nonconvex and nonsmooth, so hard to solve. In this paper, we study fundamental properties on the topology and the geometric shape of the solution set, and also conditions for convexity, connectedness, boundedness and integrality of the vertices. Further, we address various complexity issues, showing that many basic questions are NP-hard to solve. We show that the feasible set is a (nonconvex) polyhedral set and, more importantly, every nonconvex polyhedral set can be described by means of absolute value constraints. We also provide a necessary and sufficient condition when a KKT point of a nonconvex quadratic programming reformulation solves the original problem.