Convergence Of The Sequential Approximation Method For Constrained Optimization Problems

In this paper, we discuss the convergence of the sequential approximation method for con-strained optimization problems. We prove that the optimal solutions of the sequential approxi-mation problems converge to the optimal solutions of the original problem under the continuous convergency of the objective function sequence and the convergency of the constrained set se-quence. Moreover, if we assume the objective function sequence is epi-convergence instead of continuous convergence, the conclusion still holds when some monotonicity property of the ob-jective functions and the constrained sets of the sequential approximation problems is satisfied. We further apply the result to the analysis of the convergence of the smoothing methods to the complementarity constrained optimization problems.In Chapter 2, we present some background materials which will be used later on.In Chapter 3, by using the indicate function, we reformulate the constrained optimization problem to an unconstrained problem. Then we prove the epi-convergency of the objective function sequence under the continuous convergency of the objective function sequence and the convergency of the constrained set sequence. Finally, we prove that the sequence of the constrained set is eventually bounded, which leads to the convergence of the sequential approx-imation method.In Chapter 4, by using the indicate function and the maximum function, we reformulate the constrained optimization problem to an unconstrained problem. We get the convergency of the intersection of two epi-convergence sets under the epi-convergency of the objective function sequence, the convergency of the constrained set sequence and some monotonic property. We further propose a universal condition which guarantees the eventually bounded and thus the optimal solutions of the sequential approximation problems converge to the optimal solutions of the original problem.In Chapter 5, we apply the conclusion which derived in Chapter 4 to the analysis of the con-vergence of some examples including the smoothing methods of the complementarity constrained optimization problems.