Study On The Stability Theory For Nonlinear Stochastic Programming

Stochastic programming, which involves probability, classical analysis and mathematics programming, is a new mathematics branch. The stability of stochastic programming is an important direction of stochastic programming theory. Now, it has become one of the most important and active research fields of stochastic programming. Because some stochastic programs have many favorable properties and structures, it is necessary to generalize them and discuss upper semi-convergence properties of their optimal solutions set.The main method of this dissertation is to transform the convergence in distribution of discrete random variables sequence into equivalent weak convergence of probability measure sequence, so as to transform expectant functional sequence into equivalent the integration of function sequence with respect to weak convergence probability measure sequence. After further study on the integration properties of function sequence with respect to weak convergence probability measure sequence, the sequentially stochastic constrained programming is transformed into equivalent unconstrained stochastic programming. By using the epi-convergence theory, some problems on the upper semi-convergence of the approximate optimal solution set for stochastic programming expectation model, probability constrained programs model and empirical approximation model are obtained. Stability of the optimal values and the solution set of stochastic constrained programs are discussed under the framework of set-valued theory.The main contributions are listed as follows:1. Interrelations of convergence of set sequences are discussed. Continuity of probability measures on set sequences in different convergence sense is studied. Some sufficient conditions for continuous convergence of weak convergence probability measure sequence are obtained. Finally, the relation between weak convergence of probability measure and epi-convergence of probability measure is investigated, and a sufficient condition for uniform convergence of probability measure sequence is given.2. Limit theorems and dominated convergence theorems for the integration of multi-variate function sequence with respect to weak convergence probability measure sequence in the case of unbounded and semi-continuous multivariate function sequence are proved,...