Smoothing Newton algorithm for symmetric cone complementarity problems based on a one-parametric class of smoothing functions

Abstract

In this paper, we introduce a one-parametric class of smoothing functions in the context of symmetric cones which contains two symmetric perturbed smoothing functions as special cases, and show that it is coercive under suitable assumptions. Based on this class of smoothing functions, a smoothing Newton algorithm is extended to solve the complementarity problems over symmetric cones, and it is proved that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool which is extensively used in our analysis. Preliminary numerical results for randomly generated second-order cone programs and several practical second-order cone programs from the DIMACS library are reported.

A Continuation Method for Nonlinear Complementarity Problems over Symmetric Cones

“…It is also noted that there are many existing algorithms for solving NCP K (F ). These include algorithms using merit functions extended from the context of NCP [44,45], smoothing Newton methods [7,10,12,18,25,27,29,35,38], interior-point methods [39,46], and non-interior continuation methods [11,26]. All these algorithms require either the the monotonicity of F or the nonsingularity of the Jacobians of the systems involved.…”

Complementarity Problems Over Symmetric Cones: A Survey of Recent Developments in Several Aspects

“…The subjects dealt in these studies are the natural residual function [63], the Fischer-Burmeister (smoothing) function [4,59,70], Chen-Mangasarian smoothing functions [22,61,48], other merit functions [42,57,62,66,67,68,69,92,95], and smoothing continuation methods [48,61,89,21,22,126], etc.…”

A Barrier-Based Smoothing Proximal Point Algorithm for NCPs over Closed Convex Cones

“…Every solution z of the normal map equation gives a solution x = Proj K (z) of NCP K (F ); see, e.g., [16, section 1.5.2]. The normal map equation is typically solved with nonsmooth Newton's methods (see, e.g., [20] and [24]), semismooth Newton's methods (see, e.g., [13] and [17]), and smoothing Newton's methods (see, e.g., [6], [7], [8], [9], [11], [12], [19], [25], [27], [33], [44], [46], [47], and [48]). Existing nonsmooth and semismooth Newton's methods depend on the Bouligand-differentiability or the semismoothness of the nonsmooth reformulations and are thus restricted to cones whose Euclidean projector possess the same analytic properties.…”