Solving absolute value equation using complementarity and smoothing functions

Abstract

In this paper, we reformulate the NP-hard problem of the absolute value equation (AVE) as a horizontal linear complementarity one and then solve it using a smoothing technique. This approach leads to a new class of methods that are valid for general absolute value equation. An asymptotic analysis proves the convergence of our schemes and provides some interesting error estimates. This kind of error bound or estimate had never been studied for other known methods. The corresponding algorithms were tested on randomly generated problems and applications. These experiments show that, in the general case, one observes a reduction of the number of failures.

A sub-additive DC approach to the complementarity problem

“…The point x * = (1, 1) T satisfies ∇Θ(x * , F (x * )) = 0 and therefore is a stationary point of (P). However, this point is not a solution of (1).…”

“…For r sufficiently small, the functions θ approximate a step function. This observation already leads to interesting results in sparse optimization in [19] and applied in the context of complementarity problems in [1,13]. We use the function Θ to reformulate the NCP as follows…”

“…First, we present a comparison on a list of LCP problems with other DC approaches that have been suggested recently in [16]. Then, we study an adaptation of our approach to the absolute value equation and present a comparison of existing methods as in [1]. These two families of problems belong to the class of concave NCPs.…”

“…In the former, the complementarity problem (1) is reformulated as an unconstrained (non-smooth) equation solved by a Newton-kind method or an unconstrained minimization problem depending on the assumptions on F , see [3,4,6,13,14,20]. In the latter, the complementarity is ensured through a merit function reformulating (1) as an optimization problem, see [8,9,11,16]. An extensive treatment of these methods and their extensions can be found in [7].…”

“…Motivated by the recent development of DC approaches to solve the complementarity problems, we propose a new merit function based on a family of smooth sub-additive functions that has been used in the literature in the context of sparse optimization [19] and absolute value equation [1], whose latter can be reformulated as an LCP.…”