Solutions For Stochastic Variational Inequality With Conic Constraints And Applications

Variational inequality, optimization and complementarity problem with conic structure have been extensively studied in recent years. While many practical problems, such as trans-portation, finance and inventory, do not only involve deterministic data, stochastic models are proposed to reflect the uncertainties(weather or demand).The main results may be summarized as follows:1. Chapter2presents a new model which called stochastic variational inequality problem. We transform this model to a non-smooth equation system, and apply two stochastic methods to solve it. Under certain conditions, we can get convergence results. Numerical experiments show that both SA method and SAA method can effectively solve stochastic variational inequality problem.2. In Chapter3we apply sample average approximation (SAA) method based on modified Newton method to solve stochastic variational inequality with stochastic second-order cone (SSOCCVI) constraints. Under some moderate conditions, the SAA solution converges to its true counterpart with probability approaching one and convergence is exponential fast with the increase of sample size. Some illustrative examples are given to show how the globally convergent method works and comparison results between our method and other methods.3. Chapter4proposes a new class of optimization problems termed stochastic convex semi-definite programs(SCSDPs) to handle uncertain data in applications. For these model-s, we design an efficient inexact stochastic approximation(SA) method and prove the conver-gence, complexity and robust treatment of the algorithm. Apply the inexact method for solving SCSDPs where the subproblem in each iteration is only solved approximately and show that it enjoys the similar iteration complexity as the exact counterpart if the subproblems are progres-sively solved to sufficient accuracy. Numerical experiments show that the method we proposed was effective for uncertain problem.4. Chapter5apply SAA method to solve stochastic variational inequality with symmet-ric cone constraints. It provides a unified framework for dealing with the variational inequality with nonlinear constraints, variational inequality with the second-order cone constraints, and the variational inequality with semi-definite cone constraints. We apply modified Newton method based on the Fischer-Burmeister function and semi-smooth Newton method based on projec-tion function for solving SAA problem. The computation results show that the feasibility and efficiency of our algorithms.