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

Abstract

Abstract. In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [14] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer-Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [18] to the SCCP. By exploiting the Cartesian P -properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R 02 -property. All of these results generalize some recent important works in [4,25,28] under a unified framework.

A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions

“…Consequently, the first part of conclusions is a direct consequence of Proposition 4.1 of [4]. Notice that for any i ∈ O(ζ ) and υ i ∈ R n i ,…”

“…From [6], the KKT conditions for (4), which are sufficient but not necessary for optimality, can be written in the form of (1) and (2) (5) where d ∈ R n is any vector satisfying Ax = b. For large problems with a sparse A, (5) has an advantage that the main cost of evaluating the Jacobian ∇F and ∇G lies in inverting AA T , which can be done efficiently via sparse Cholesky factorization.…”

“…The class of functions is a natural extension of the family of NCP functions proposed by Kanzow and Kleinmichel [17], and has been shown in [4] to satisfy the characterization (6). It is not hard to see that as τ = 2, φ τ reduces to the FB function φ FB in (7) while it becomes a multiple of the natural residual function φ NR as τ → 0 + .…”

“…In [4], we studied the continuous differentiability of ψ τ and showed that each stationary point of τ is a solution of the SOCCP if ∇F and −∇G are column monotone. In this paper, we concentrate on the properties of φ τ , including the globally Lipschitz continuity, the strong semismoothness, and the characterization of the B-subdifferential.…”

“…In this paper, we concentrate on the properties of φ τ , including the globally Lipschitz continuity, the strong semismoothness, and the characterization of the B-subdifferential. Particularly, we provide a weaker condition than [4] for each stationary point of τ to be a solution of the SOCCP and establish the boundedness of the level sets of τ , by using Cartesian P -properties. We also propose a semismooth Newton method based on the system (10), and obtain the corresponding global and the superlinear convergence results.…”