COAP 2007 Best Paper Award

Abstract

Each year, the Computational Optimization and Applications (COAP) editorial board selects a paper from the preceding year’s COAP publications for the Best Paper Award. The recipient of the award for papers published in 2007 is Olvi Mangasarian of the University of Wisconsin, Madison and the University of California, San Diego, for his paper “Absolute Value Programming”, published in Volume 36, pages 43–53. This paper [7] as well as subsequent closely related papers [6, 8, 9] deal with the absolute value equation (AVE) Ax + B|x| = b, where A and B are arbitrary m× n real matrices and |x| denotes the vector with absolute values of the n-dimensional real valued vector x. The significance of this class of NP-hard problems arises partly from the fact that when B = I , the identity matrix, AVE is equivalent to the general linear complementarity problem, 0 ≤ x ⊥Mx + q ≥ 0. Even though problems involving absolute values are NP-hard, they share some very interesting properties with those of linear systems. For example, optimization problems with absolute value constraints possess optimality and duality results similar to those of linear programming, even though the problems are inherently nonconvex. Another interesting property that AVE shares with linear inequalities are theorems of the alternative which are established in this paper. The paper also contains a finite successive linearization algorithm for solving absolute value equations that terminates at a necessary optimality condition. This algorithm has solved all random test problems given to it for which mostly m ≥ 2n or n ≥ 2m, up to size (m,n) of (2000,100) and (100,2000). When m = n and B is invertible, which is the case for the linear complementarity problem formulation, a simpler concave minimization