Class Of Merit Functions Research Articles

Three classes of merit functions for the complementarity problem over a closed convex cone

In this article, we consider the complementarity problem over a closed convex cone in a Hilbert space. We propose three classes of merit functions for solving such a problem, including natural residual merit functions, LT merit functions and implicit Lagrangian merit functions, and investigate their important properties related to the unconstrained reformation of the problem, including coerciveness properties of functions and the related error bound results. In addition, we also propose a necessary and sufficient condition of a stationary point of the unconstrained reformation being a solution to the original problem for implicit Lagrangian merit functions. These provide a theoretical basis for designing the merit function method for solving the problem concerned.

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

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.

Lipschitz continuity of the gradient of a one-parametric class of SOC merit functions

In this article, we show that a one-parametric class of SOC merit functions has a Lipschitz continuous gradient; and moreover, the Lipschitz constant is related to the parameter in this class of SOC merit functions. This fact will lay a building block when the merit function approach as well as the Newton-type method are employed for solving the second-order cone complementarity problem with this class of merit functions.

Two Classes of Merit Functions for Infinite-Dimensional Second Order Complimentary Problems

In this article, we extend two classes of merit functions for the second-order complementarity problem (SOCP) to infinite-dimensional SOCP. These two classes of merit functions include several popular merit functions, which are used in nonlinear complementarity problem, (NCP)/(SDCP) semidefinite complementarity problem, and SOCP, as special cases. We give conditions under which the infinite-dimensional SOCP has a unique solution and show that all these merit functions provide an error bound for infinite-dimensional SOCP and have bounded level sets. These results are very useful for designing solution methods for infinite-dimensional SOCP.

Coerciveness of some merit functions over symmetric cones

Complementarity problems over symmetric cones (SCCP) can be reformulated as the global minimization of a certain merit function. The coerciveness of the merit function plays an important role in this class of methods. In this paper, we introduce a class of merit functions which contains the Fischer-Burmeister merit function and the natural residual merit function as special cases, and prove the coerciveness of this class of merit functions under some conditions which are strictly weaker than the assumption that the function involving in the SCCP is strongly monotone and Lipschitz continuous. Based on the introduced merit function, we propose another class of merit functions which is an extension of Fukushima-Yamashita merit function. We investigate the coerciveness of the generalized Fukushima-Yamashita merit function under a condition which is strictly weaker than the assumption that the function involving in the SCCP is weakly coercive. The theory of Euclidean Jordan algebras is a basic tool in our analysis.

On Minimizing and Stationary Sequences of a New Class of Merit Functions for Nonlinear Complementarity Problems

Motivated by the work of Fukushima and Pang (Ref. 1), we study the equivalent relationship between minimizing and stationary sequences of a new class of merit functions for nonlinear complementarity problems (NCP). These merit functions generalize that obtained via the squared Fischer–Burmeister NCP function, which was used in Ref. 1. We show that a stationary sequence {xk} ⊂ /Ren is a minimizing sequence under the condition that the function value sequence {F(xk)} is bounded above or the Jacobian matrix sequence {F′(xk)} is bounded, where F is the function involved in NCP. The latter condition is also assumed by Fukushima and Pang. The converse is true under the assumption of {F′(xk)} bounded. As an example shows, even for a bounded function F, the boundedness of the sequence {F′(xk)} is necessary for a minimizing sequence to be a stationary sequence.