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

Abstract

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.

Keywords:

• complementarity problem
• closed convex cone
• merit function
• coerciveness
• error bound
• stationary point

AMS Subject Classifications::

• 90C22
• 90C25
• 90C33

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant No. 10871144).