The CVaR-SP Reformulation And The DRO Reformulation For Stochastic Complementarity Problems

The complementarity problem is one of the important topics in mathematical programming. And it has a wide range of applications in the fields of engineering and science, such as control theory, operations research, finance and so on. However, in many practical issues, the parameters in complementarity problems involve uncertainty. Thus, it is much essential to study the stochastic complementarity problem with uncertain parameters. Being affected by the uncertain parameters, the stochastic complementarity problem may have no solution. It is one of the vital research directions in the stochastic complementarity problem to establish an reasonable deterministic approximation, which is also our research interests. The main contributions of this dissertation are as follows:1. We investigate the stochastic nonlinear complementarity problem, and present a CVaR-constrained stochastic programming (CVaR-SP) reformulation, which minimizes the expected residual defined by a restricted NCP function with nonnegative constraints and conditional value-at-risk (CVaR) constraints, which are used to approximated the chance constraints that guarantee the stochastic nonlinear function being nonnegativity with a probability not less than the given high level. By applying smoothing technique and penal-ty method, we propose a penalized smoothing sample average approximation (SAA) method to solve the CVaR-constrained stochastic programming. We show the almost surely con-vergence of the optimal solutions of the penalized smoothing sample average approximation problem to the solution of the corresponding nonsmooth CVaR-constrained stochastic pro-gramming. Numerical test shows this reformulation and the smoothing penalty SAA method are efficient.2. We investigate the stochastic linear complementarity problem (SLCP) affinely affect-ed by the uncertain parameters. Assuming that the distribution of the uncertain parameters belongs to some ambiguity set with prescribed partial information, such as the information about the first two moments or together with additional information about the support, we formulate the SLCP as a distributionally robust optimization (DRO) reformulation, in terms of a constrained minimization problem, where the objective function is the worst-case of an expected complementarity measure and the constraints contain nonnegativity constraints and a worst-case joint chance constraint that the linear mapping is nonnegative with a probability not less than the given level. Applying the cone dual theory and S-procedure, we show that the DRO reformulation can be conservatively approximated by an nonlinear semidefinite programming (NSDP) with bilinear matrix inequalities or a NSDP with a non-linear objective and linear matrix inequalities, which can be solved by the PENLAB solver for NSDPs. Preliminary numerical results show that the DRO reformulation as well as the corresponding method is desirable.3. We investigate the SLCP affected by uncertain parameters affinely or quadratically. Assuming that the distribution of the uncertain parameters belongs to some ambiguity set with prescribed partial information, such as the information about the first two moments or additional information about the support, we formulate SLCP as a more reasonable dis-tributionally robust optimization (RDRO) reformulation, which contains only subproblem with respect to the distribution. By considering the worst-case of the expected complemen-tarity measure with a joint chance constraint that the linear mapping is nonnegative with a probability not less than the given level as a subproblem with respect to the distribution, the RDRO reformulation minimizes the worst-case of the optimal value of the subproblem with nonnegative constraints. Applying the cone dual theory and S-procedure, we conserva-tively approximate the RDRO by an NSDP with bilinear matrix inequalities, which can be solved by the PENLAB solver for NSDPs. The preliminary numerical test on a constrained stochastic linear quadratic control problem shows that the RDRO reformulation as well as the corresponding solution method is promising.