The Theory And Algorithms For Nonmonotone Symmetric Cone And Homogeneous Cone Complementarity Problems

The study of symmetric cone complementarity problems has been a hottopic in the literature since the 1990's. The model provides a simple, natu-ral and unified framework for various existing complementarity problems suchas the standard complementarity problem, the second-order cone complemen-tarity problem and the semidefinite complementarity problem. It has wideapplications in engineering, economics, management science and other fields.As an extension of symmetric cone complementarity problems, homogeneouscone complementarity problems have been studied extensively. This thesis ismainly concerned with nonmonotone symmetric cone complementarity prob-lems and nonmonotone homogeneous cone complementarity problems from thetwo aspects of the theory and algorithm by using the techniques of EuclideanJordan algebras and T-algebras.Firstly, we do two works of research in algorithms for nonmonotone sym-metric cone complementarity problems. A smoothing algorithm is extended tosolve the linear complementarity problem over Euclidean Jordan algebras witha Cartesian P_*(κ)-transformation. We show that the algorithm is globally con-vergent if the problem concerned has a solution. In particular, we show thatthe algorithm is globally linearly convergent under a new assumption which isweaker than most assumptions used in smoothing algorithms. On the otherhand, a smoothing Newton algorithm is also extended to solve the symmet-ric cone linear complementarity problem with a Cartesian P0-transformation.By using the theory of Euclidean Jordan algebras, we show that the systemof Newton equations is solvable and the transformation H(μ,x,s) is coercivein (x,s) which ensures that the level set is bounded. By these two results,we further show that the algorithm is well-defined and globally and locallyquadratically convergent under some assumptions which are weaker than thosein the literature. Secondly, for the relaxation transformation Rφover a Euclidean Jordanalgebra introduced by Tao and Gowda, we further investigate some intercon-nections between properties ofφand properties of Rφ, including the propertiesof continuity, (local) Lipschitz continuity, directional di_*erentiability, (contin-uous) differentiability, semismoothness, monotonicity, the P0-property, and theuniform P-property. These properties provide a theoretical foundation for thefurther research of nonmonotone symmetric cone complementarity problems.Finally, we introduce the concepts of w-P and w-uniqueness propertiesfor nonlinear transformations defined on T-algebras and study some intercon-nections between these concepts. We also specialize them to relaxation trans-formations and self-adjoint linear transformations. Furthermore, we study thefiniteness of w-solutions for the homogeneous cone complementarity problem.