Vector-Valued Implicit Lagrangian for Symmetric Cone Complementarity Problems

Abstract

The implicit Lagrangian was first proposed by Mangasarian and Solodov as a smooth merit function for the nonnegative orthant complementarity problem. It has attracted much attention in the past ten years because of its utility in reformulating complementarity problems as unconstrained minimization problems. In this paper, exploiting the Jordan-algebraic structure, we extend it to the vector-valued implicit Lagrangian for symmetric cone complementary problem (SCCP), and show that it is a continuously differentiable complementarity function for SCCP and whose Jacobian is strongly semismooth. As an application, we develop the real-valued implicit Lagrangian and the corresponding smooth merit function for SCCP, and give a necessary and sufficient condition for the stationary point of the merit function to be a solution of SCCP. Finally, we show that this merit function can provide a global error bound for SCCP with the uniform Cartesian P-property.

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

“…In this section, we show that the regularized merit function f τ can provide a global error bound for the solution of the SCCP under a different condition from [21]. To the end, we need the concepts of Cartesian P -properties introduced in [17] for a nonlinear transformation, which are natural extensions of the P -properties on Cartesian products in R n established by Facchinei and Pang [11] and the Cartesian P -properties in the setting of S n developed by Chen and Qi [6]. …”

“…It deserves further investigation. To the contrast, inequality (63) can be easily verified for the Implicit Lagrangian merit function in SCCP case (see [17]), and conditions for each stationary point being a solution of the SCCP is also established therein. However, the analysis can not be carried over to ψ τ due to the more complicated structure of ∇ψ τ .…”

“…Kong, Tuncel and Xiu [17] studied the extension of the implicit Lagrangian function proposed by Mangasarian and Solodov [22] to symmetric cones; and Kong, Sun and Xiu [16] proposed a regularized smoothing method for the SCCP (3) based on the natural residual complementarity function associated with symmetric cones. Following this line, in this paper we will consider the extension of the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [14] and a class of regularized functions based on it.…”

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.…”

Convergence of a smoothing algorithm for symmetric cone complementarity problems with a nonmonotone line search

“…The main focus has been on extending various interiorpoint algorithms to solving optimization problems over symmetric cones [11−15] . Recently, some smoothing algorithms have been proposed for solving the linear programs over symmetric cones [16] and the symmetric cone complementarity problems [17,18] ; and some merit function algorithms have been proposed for the symmetric cone complementarity problems [19,20] . It is easy to show that the assumptions used in literature mentioned above imply that the solution set of the problem concerned is nonempty and bounded.…”