A Smoothing Approach For Mathematical Programs With Symmetric Cone Complementarity Constraints And Applications

This thesis discusses a class of mathematical programs with symmetric cone complementarity constraints (SCMPCCs), which is a special case of mathematical programs with equilibrium problems (MPECs) and has applications not only in engineering design and economics but also in many mathematical problems, such as symmetric cone bilevel programs, robust optimizations and inverse problem over symmetric cones. Therefore, it is of great important to study the optimality theories and numerical methods of this type of problems. This thesis focuses on the stationary conditions and smoothing approaches for the mathematical program with symmetric cone complementarity constraints. Two special cases, the mathematical program with second-order cone complementarity constraints (SOCMPCC) and the mathematical program with semidefinite cone complementarity constraints (SDCMPCC), are discussed in detail.The main results of this dissertation can be summarized as follows:In Chapter3, the stationary condition and linear independence constraint qualifications for the mathematical program with symmetric cone complementarity constraints are introduced. With the help of the general Jacobian of the projection operator on a symmetric cone, the C-stationary condition and SCMPCC-linear independence constraint qualification (denoted by SCMPCC-LICQ) are first presented. For its special case-the mathematical program with second-order cone complementarity constraints, we not only introduce the C-stationary points by the generalized Jacobian of the projection operator onto a second-order cone, but also present the M-, S-stationary points based on the regular and limiting coderivatives of the projection operator. Meanwhile, by the definitions of the SOCMPCC-linear independence constraint qualification (denoted by SOCMPCC-LICQ) and second-order cone upper level strict complementarity (denoted by SOC-ULSC) condition, the relations among these stationary points are established.Chapter4focuses on the convergence analysis of the smoothing approach for the mathematical program with symmetric cone complementarity constraints. We employ the smoothing projection operator over a symmetric cone to approximate the symmetric cone complementarity constraints and construct a smoothing approximation of this problem. The convergence behavior of the feasible set, optimal solution set, optimal value and stationary points of the smoothing approximated problem is discussed when the smoothing parameter tends to zero. In particular, it is shown that the convergence rate of the Hausdorff distance between the feasible set of the smoothing approximated problem and the feasible set of the origin problem is of the same order with the smoothing parameter, the optimal solution set and optimal value of the smoothing approximated problem are outer semicontinuous and Lipschitz continuous as the smoothing parameter tends to zero, respectively. Moreover, stationary points of the smoothing approximated problem converge to a C-stationary point under SCMPCC-LICQ. For the smoothing approaches for the special cases-the mathematical program with second-order cone complementarity constraints and the mathematical program with semidefinite cone complementarity constraints, they inherit all the convergence properties of the smoothing approach for mathematical programs with symmetric cone complementarity constraints. Furthermore, we demonstrate that the stationary points of the smoothing approximated problems are also convergent to M-, S-stationary points of these two problems under different conditions respectively.Chapter5is devoted to studying a type of inverse second-order cone quadratic programming problems, in which the parameters in both the objective function and the constraint set of a given second-order cone quadratic programming problem need to be adjusted. This inverse problem can be reformulated as a linear second-order cone complementarity constrained optimization problem with a semismoothly differentiable objective function, which is essentially a mathematical program with second-order cone complementarity constraints. A smoothing approximated problem is constructed with the help of the smoothing projection operator over a second-order cone, whose feasible set and optimal solution set are demonstrated to be continuous and outer semicontinuous respectively as the parameter decreases to zero. A smoothing Newton method is constructed to solve the smoothing problem and its global convergence and local quadratic convergence rate are shown. Finally, the numerical results implemented by Matlab codes are reported to show the effectiveness for the smoothing Newton method to solve the inverse second-order cone quadratic programming problem.