A Continuation Method for Nonlinear Complementarity Problems over Symmetric Cones

Abstract

In this paper, we introduce a new P -type property for nonlinear functions defined over Euclidean Jordan algebras, and study a continuation method for nonlinear complementarity problems over symmetric cones. This new P -type property represents a new class of nonmonotone nonlinear complementarity problems that can be solved numerically.

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.…”

“…+ is called the normal map of the SCCP and known as a C-function for the SCCP (see [21] and also see Section 1.5 of [25] in the case of the NCP). In contrast, the above properties are different for the SCCP in general.…”

“…Furthermore, Chua, Lin and Yi [22] (see also [21]) introduced the following property, uniform nonsigularity, of the nonlinear operator ψ : V → V in NLCP(ψ, q), …”

“…Note that the uniform nonsigularity is an extension of P-properties, i.e., if V = n the uniform nonsigularity is equivalent to P-function property (see Proposition 4.1 of [21]). In [21], a Chen and Mangasarian smoothing method has been proposed for solving NLCP(ψ, q) where ψ is uniformly nonsingular.…”

“…In [21], a Chen and Mangasarian smoothing method has been proposed for solving NLCP(ψ, q) where ψ is uniformly nonsingular. In [22], the authors showed several implications among the Cartesian P properties and uniform nonsigularity and discussed the existence of Newton directions and the boundedness of iterates of some merit function methods. The authors also showed that LCP(L, q) is globally uniquely solvable under the assumption of uniform nonsigularity.…”