Some Studies On Stochastic Mathematical Programs With Equilibrium Constraints

Mathematical programs with equilibrium constraints (MPEC) have found extensive applications in a number of areas such as economic equilibrium, game theory and engi-neering design. Over the past few decades, MPECs have been extensively studied from optimization theory to numerical methods. In practice, MPEC models often involve uncer-tainty when they are applied to describe decision making problems which involve decision makers with hierarchical relationships and future uncertainty. Consequently, stochastic versions of MPEC (SMPEC) models are needed.In the past decade, people have proposed two kinds of SMPEC models. One is called Here-and-Now SMPEC and, the other is called lower-level Wait-and-See SMPEC. In this paper, we study SMPECs on theory, algorithms and models.In [54], Meng and Xu proposed a regularized sample approximation (SAA) method for solving the Here-and-Now type SMPEC. However, they have not presented any con-vergence analysis. In Chapter 3, we fill this gap. We study the convergence properties of the optimal solutions and optimal values. We also show that, under some moderate con-ditions, the stationary points of the SAA-regularized problems converge to a C-stationary (M-stationary or B-stationary) point of the original SMPEC with probability one (w.p.1).In Chapter 4, we propose a partial penalized sample average approximation method to optimize the Here-and-Now type SMPEC. We prove that w.p.1 the optimal solutions and stationary points converge to the counterparts of the true SMPEC. Exponential rate of convergence of estimators is also established under some additional conditions. For the convergence results although we only require the MPEC-NNAMCQ for the original problem and hence MPEC-LICQ may fail for the original problem (4.1), the MPEC-LICQ is satisfied at every feasible point of the penalized problem regardless of structure of the original problem. From a numerical perspective, this is very important as the stability of many existing numerical methods depend on MPEC-LICQ. Compared with [8,43,54], the proposed method needs relatively weaker conditions to guarantee the convergence. We also provide 5 numerical tests on the proposed method. Since the feasible sets of the SAA-penalized problems do not depend on the sample size, it may be easier for programming.In Chapter 5, we present numerical approximation schemes for the lower-level Wait-and-See type SMPEC. By treating the approximation problems as the perturbations of the original (true) problem, we carry out a detailed stability analysis of the approximated problems including continuity and local Lipschitz continuity of optimal value functions, and outer semicontinuity and continuity of the set of optimal solutions and stationary points. Difference from [45,78,90], we do not assume that the complementarity con-straints have unique solution. At the same time, we study quantitative stability of the solution sets, optimal value and stationary points of the SMPEC underlying probability measure varies in some metric probability space. To our knowledge, it is the first time to study the stability of SMPEC with respect to probability measure. A particular focus is given to empirical probability measure approximation which is also known as sample average approximation (SAA). It is shown that the optimal values and stationary points of approximated problems converge to the optimal value and M-(C-, S-) stationary point of the original SMPEC.In Chapter 6, we propose a new SMPEC model, Mathematical program with hy-brid equilibrium constraint (SMPHEC). The new model is a combination of relatively well studied Here-and-Now type SMPEC and lower level Wait-and-See type SMPEC. An example is given to describe the necessity to study SMPHEC. By assuming that the per scenario complementarity system is strongly monotone, the SMPHEC is converted into a Here-and-Now type SMPEC by virtue of a nonsmooth implicit function theorem. A smoothing penalized sample approximation method is proposed to solve the problem. Asymptotic consistency of statistical estimators of optimal solutions and stationary points are demonstrated under some moderate conditions.