A Smoothing Method with Appropriate Parameter Control Based on Fischer-Burmeister Function for Second-Order Cone Complementarity Problems

Yasushi Narushima,Hideho Ogasawara,Shunsuke Hayashi

doi:10.1155/2013/830698

Abstract

We deal with complementarity problems over second-order cones. The complementarity problem is an important class of problems in the real world and involves many optimization problems. The complementarity problem can be reformulated as a nonsmooth system of equations. Based on the smoothed Fischer-Burmeister function, we construct a smoothing Newton method for solving such a nonsmooth system. The proposed method controls a smoothing parameter appropriately. We show the global and quadratic convergence of the method. Finally, some numerical results are given.

Highlights

We review some propositions needed to establish convergence properties of the smoothing Newton method
The function HFB is locally Lipschitzian on R2n+l and, is semismooth on R2n+l
Since F is a continuously differentiable function, HFB is locally Lipschitzian on R2n+l

Summary

Introduction

We consider the second-order cone complementarity problem (SOCCP) of the following form: find (x, y, p) ∈ Rn × Rn × Rl such that (1). Fukushima et al [5] showed that the natural residual function, called the min function, and the Fischer-Burmeister function for the NCP can be extended to the SOCCP by using the Jordan algebra They further constructed the smoothing functions for those SOC complementarity (C-) functions and analyzed the properties of their Jacobian matrices. Chen et al [7] proposed another smoothing method with the natural residual in which the smoothing parameter is treated as a variable in contrast to [6] They showed the Abstract and Applied Analysis global and quadratic convergence of their method. Similar to Chen et al, Narushima et al [8] proposed a smoothing method treating a smoothing parameter as a variable They used the Fischer-Burmeister function instead of the natural residual function and provided the global and quadratic convergence of the method. For a given set S ⊂ Rn, int S, bd S, and conv S mean the interior, the boundary, and the convex hull of S in Rn, respectively

Some Preliminaries

Smoothing Newton Method and Its Convergence Properties

Numerical Experiments

Conclusion