Some Studies On Mathematical Programs With Second-order Cone Complementarity Constraints And Mathematical Programs With Equilibrium Constraints

Mathematical Program with Second-Order Cone Complementarity Constraints (abbreviated as MPSOCC) is a constrained optimization problem in which the essen-tial constraints include second-order cone complementarity systems. One of the main sources of MPSOCC comes from-a class of Bilevel Programming Problem (abbreviat-ed as BLPP) which contains a convex Second-Order Cone Program (abbreviated as SOCP) or Robust Optimization Problem (abbreviated as ROP) as lower level pro-gram, it can be formulated as an MPSOCC by replacing the SOCP or ROP by its KKT condition. Mathematical Programs with Equilibrium Constraints (abbreviated as MPEC) can be regarded as a special case of the MPSOCC when the Second-Order Cone reduces to the set of nonnegative reals. MPSOCC and MPEC plays a very important role in many fields such as economic equilibria, transportation science and engineering design.First of all, motivated by the MPCC theory and method, the Clarke-stationarity concept be presented in this paper. In addition, we consider the Strong stationarity condition. Moreover, we present two approximation smoothing methods and a relax-ation method for solving the MPSOCC. The last, we present a modified Levenberg-Marquardt method for solving the MPEC. The main results of this dissertation can be summarized as follows:1. In Chapter3, we first give an MPSOCC variant of calmness, called MPSOCC-calm, motivated by the calmness condition for nonlinear programming problem. And we give the first order necessary optimality conditions for MPSOCC under the notion of the Clark subdifferential. It is further shown that a Iocal minimizer of MPSOCO must be Clark-stationary in the condition of MPSOCC-calm. In addition we give the MPSOCC-strict nondegeneracy, the condition extends the nondegenerate condition in SOCPs. We consider the Strong stationarity condition, it is further shown that a local minimizer of MPSOCC must be Strong-stationary point in the condition of MPSOCC-strict nondegeneracy.2. In Chapter4, we first present two approximation smoothing methods for solving the MPSOCC. We formulated the MPSOCC as an nonlinear programming problem by replacing the SOCP by the natural residual function and Fischer-Burmeister function respectively. We show that, under the MPSOCC-strict nondegenerate, any accumu-lation point of stationary points of the approximation problems must be a Clarke-type stationary point of the MPSOCC. In addition we present a relaxation method for solving the MPSOCC. We show that, under the MPSOCC-strict nondegenerate, any accumulation point of stationary points of the approximation problem must be a Clarke-type stationary point of the MPSOCC. We also show that Lagrange multipliers of the approximation problem exist under the MPSOCC-strict nondegenerate.3. In Chapter5, we aims at developing effective numerical methods for solving mathematical programs with equilibrium constraints. We reformulate these station-arity conditions as nonlinear equations. And then present a modified Levenberg-Marquardt method for solving these nonlinear equations. We show that, under some weak local error bound conditions, the method is locally and superlinearly convergent. Furthermore, the numerical results implemented by Matlab codes are reported to show the effectiveness for this method to solve the MPEC.