Approximate Proximal Point Methods For Minimizing The Sum Of Two Convex Functions

Recently, many researchers focus on minimizing the structured optimization problem where the objective function is the sum of two convex functions. Alternating linearization method can be viewed as a class of approximate proximal point methods and is one of the e?ective methods for solving the problem. The objective function is linearized alternately by alternating linearization method, so the original problem is transformed into a sequence of regularized subproblems. But if the function with highly nonlinearity is replaced by the linear model, then the caticulation error may be too large. The linear model in the alternating linearization method is obtained by linearization at the current point. This paper improves the linear model by taking account of the existing iterative points. The main results may be summarized as follows:Chapter 2 proposes a new proximal point method. The function with less nonlinearity is replaced by the linear model and the other is replaced by its piecewise linear approximation model. Convergence results are established, and numerical examples are given to show the e?ciency of the method. Finally the dual application of the method is presented.Chapter 3 solves a class bilevel programming problems using the new proximal point method. The convergence results are given and some numerical examples are presented to illustrate the e?ciency of the algorithm.Chapter 4 presents a new method of alternating linearization method. The function is replaced by the linear model and the other is replaced by the convex combination of linear model of current point and linear model of previous point. Finally we solves a class bilevel programming problems using this method. we also give some numerical examples to show the e?ciency of the method.