A Two-Parametric Class of Merit Functions for the Second-Order Cone Complementarity Problem

Xiaoni Chi,Zijun Hao,Zhongping Wan

doi:10.1155/2013/571927

Abstract

We propose a two-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) based on the one-parametric class of complementarity functions. By the new class of merit functions, the SOCCP can be reformulated as an unconstrained minimization problem. The new class of merit functions is shown to possess some favorable properties. In particular, it provides a global error bound ifFandGhave the joint uniform CartesianP-property. And it has bounded level sets under a weaker condition than the most available conditions. Some preliminary numerical results for solving the SOCCPs show the effectiveness of the merit function method via the new class of merit functions.

Highlights

We consider the following second-order cone complementarity problem (SOCCP) of finding (x, y, ζ) ∈ Rn × Rn × Rn such that⟨x, y⟩ = 0, x ∈ K, y ∈ K, (1)x = F (ζ), y = G (ζ), where ⟨⋅, ⋅⟩ is the Euclidean inner product and F : Rn → Rn and G : Rn → Rn are continuously differentiable mappings
We propose a two-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) based on the one-parametric class of complementarity functions
We aim to study the twoparametric class of merit functions for the SOCCP defined as (7)–(10) and its favorable properties in this paper

Summary

Introduction

We consider the following second-order cone complementarity problem (SOCCP) of finding (x, y, ζ) ∈ Rn × Rn × Rn such that. The SOCCP contains a wide class of problems, such as nonlinear complementarity problems (NCP) and second-order cone programming (SOCP) [3, 4]. There is an alternative approach [14, 15] based on reformulating the SOCCP as an unconstrained smooth minimization problem. In that approach, it aims to find a smooth function ψ : Rn × Rn → R+ such that ψ (x, y) = 0 ⇐⇒ ⟨x, y⟩ = 0, x ∈ K, y ∈ K.

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