Some Properties of a Class of Merit Functions for Symmetric Cone Complementarity Problems

Abstract

In this paper, we extend a class of merit functions proposed by Kanzow et al. (1997) for linear/nonlinear complementarity problems to Symmetric Cone Complementarity Problems (SCCP). We show that these functions have several interesting properties, and establish a global error bound for the solution to the SCCP as well as the level boundedness of every merit function under some mild assumptions. Moreover, several functions are demonstrated to enjoy these properties.

A one-parametric class of merit functions for the second-order cone complementarity problem

“…In Section 3 and Section 4, we show that the function ψ τ defined by (6) is continuously differentiable everywhere and has a globally Lipschitz continuous gradient with the Lipschitz constant being a positive multiple of 1 + τ −1 . These results generalize some recent important works in [4,25,28] under a unified framework, as well as improve the work [21] greatly in which only the differentiability of the merit function ψ FB was given.…”

“…Recently, motivated by the successful applications of the merit function approach in the solution of NCPs, SOCCPs and SDCPs (see, e.g., [4,10,22,28]), some researchers started with the investigation of merit functions or complementarity functions associated with symmetric cones. For example, Liu, Zhang and Wang [21] extended a class of merit functions proposed in [18] to the following special SCCP:…”

“…Using the properties of ψ 0 in (9), it is not hard to verify that f τ is a merit function for the SCCP. The class of functions will reduce to the one studied in [21] if τ = 2 and G degenerates into an identity transformation. In Section 5, we show that the class of merit functions can provide a global error bound for the solution of the SCCP under the condition that G and F have the joint uniform Cartesian P -property.…”

“…In Section 5, we show that the class of merit functions can provide a global error bound for the solution of the SCCP under the condition that G and F have the joint uniform Cartesian P -property. In Section 6, we establish the boundedness of the level sets of f τ under a weaker condition than the one used by [21], which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with G and F having the joint Cartesian R 02 -property.…”

Monotonicity of Löwner operators and its applications to symmetric cone complementarity problems

“…SCCP provides a simple, natural, and unified framework for various existing complementarity problems, such as the NCP, SDCP and the second-order cone complementarity problem (SOCCP); see, e.g., [3,5,6,8,11,14,21].…”