Differential Properties Of SAA Solution Mappings Governed By Parametric Stochastic Generalized Equations And Applications

As uncertain elements exist in many practical problems, the stochastic program prob-lems, especially the stochastic equilibrium problems have become an important concern in recent years. One of the key points in the study of stochastic equilibrium problems lies in the differential properties for the solution mapping governed by parametric stochastic generalized equations. This dissertation focuses on the differential properties of SAA solu-tion mappings to parametric stochastic generalized equations and applications, including convergence analysis of coderivatives of SAA solution mapping to a parametric stochastic generalized equation, convergence analysis of coderivatives of SAA solution map to a con-strained parametric stochastic variational inequalities and a smoothing SAA method for a stochastic mathematical program with complementarity constraints. The main results of this dissertation can be summarized as follows:1. In Chapter 3, convergence of coderivative of the SAA solution mapping to a para-metric stochastic generalized equation is studied. Because of the unboundedness of coderivative, the notions of cosmic deviation and the integrated deviation are introduced to estimate the deviation between unbounded sets. It is demonstrated that, under suitable conditions, both the cosmic deviation and the integrated devi-ation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic generalized equation converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution mappings to the SAA parametric generalized equations is established. At last, the results are used to analyze the con-sistency of the Lipschitz-like property (or Aubin property) of the solution mapping of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic mathematical program with complementarity constraints.2. Chapter 4 focuses on the convergence analysis of coderivative of SAA solution mapping to a stochastic equality and inequality constrained parametric stochastic variational inequality. Although the model studied in this chapter is a special case of para-metric stochastic generalized equations, the SAA method is different from the one in Chapter 3 because the expectations in its constraints also need to be approx-imated by the sample averaging approach. Under suitable conditions, we at first show that the coderivative of the solution mapping to SAA problem exponentially converges to the coderivative of solution mapping to the parametric stochastic gen-eralized equation as the sample size tends to infinity with probability 1. And then, the results are used to analyze the consistency of the Lipschitz-like property of the solution mapping of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic bilevel program.3. In Chapter 5, a smoothing SAA method for a stochastic mathematical program with complementarity constraints (SMPCC) problem is discussed. Under suitable con-ditions, the almost sure convergence of the optimal solutions of the smoothed SAA problem is characterized by the notion of epi-convergence in variational analysis. It is also demonstrated that any accumulation point of Karash-Kuhn-Tucker points of the smoothed SAA problem is almost surely a kind of stationary point of SM-PCC as the sample size tends to infinity. At last, under a strong second-order sufficient condition for SMPCC, the exponential convergence rate of the sequence of Karash-Kuhn-Tucker points of the smoothed SAA problem is obtained through an application of Robinson's stability theory. Some preliminary numerical results are reported to show the efficiency of the method proposed.