C-Functions And Merit Functions For Symmetric Cone Complementarity Problems

Symmetric cone complementarity problem (SCCP) includes the nonnegative orthant nonlinear complementarity problem (NCP), the second-order cone complementarity problem (SOCCP), and the semi-definite complementarity problem (SDCP). This model provides a simple, natural, and unified framework. It has wide applications in engineering, economics, management science, and other fields. This thesis is mainly concerned with complementarity functions (C-functions) and merit functions for SCCP.Jordan algebraic technique provides a useful tool for describing and analyzing symmetric cone optimization problem, and the Jordan frame plays an important role in Euclidean Jordan algebras. In Chapter 1, recalling some concepts and results on Euclidean Jordan algebras, we give necessary and sufficient conditions under which there is a unique Jordan frame in a Euclidean Jordan algebra.In Chapter 2, employing the Jordan-algebraic structure, we extend the implicit function by Mangasarian and Solodov for NCP to the vector-valued implicit Lagrangian for SCCP, and show that it is a continuously differentiable and strongly semismooth C-function for SCCP. 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. We also show that this merit function can provide a global error bound for SCCP with the uniform Cartesian P-property. Finally, a hybrid Newton method is established, which is quadratically convergent under appropriate assumptions.Chapter 3 reviews the Lowner operator and develops some new results on the Jacobian operator. Then we study the strong semismoothness and Jacobian nonsingularity of a natural residual function for SCCP, and investigate the level-boundedness of the related merit functions for SCCP under certain monotonic-ity conditions. Based on the uniform approximation property and Jacobian consistency of the Chen-Mangasarian class of regularized smoothing functions, we develop a globally and quadratically convergent algorithm for solving monotone SCCPs. In chapter 4, we establish EP classes and Mangasarian class of C-functions which solves an open question by Tseng in 1998. We also show that the former are continuously differentiable and strongly semismooth everywhere. Moreover, we mainly show necessary and sufficient conditions for locally Lipschitz Lowner operator to be monotone, strictly monotone and strongly monotone.