A Hybrid Approximate Proximal Point Method For Minimizing The Sum Of Two Convex Functions

Hybrid Approximate Proximal Point Method(HAPPM), which is an efective methodfor minimizing the sum of two convex functions, especially for the function which has highnonlinearity, is proposed on the basis of the Alternating Linearization Method(ALM). Itreplaces the optimization problem in the proximal point algorithm by a series of subprob-lems of minimizing the approximate functions to get the optimal solution of optimizationproblem. In the subproblems, the function with less nonlinearity is replaced by its linearmodel and the other is replaced by its quadratic model alternately. Under the frameworkof the proximal point algorithm, we can find the solution of the original problem.The main work of this paper is to propose a new method, named Hybrid ApproximateProximal Point Method. The idea is that the function with less nonlinearity is replaced byits linear model and the other is replaced by its quadratic model alternately. In this paper,it analyzes the HAPPM from two aspects of theory and numerical example. Theoretically,it is global convergence can be established under certain conditions. In finally, numericalexamples show that the method is efective.The present thesis is organized as follows. In Chapter1, we firstly give a brief intro-duction on the development of some decomposition method commonly used in nonlinearprogramming and then give some related concepts and notations. The second chapter andthe third chapter introduce basic results of Proximal Point Algorithm(PPA) and ALM,respectively. In Chapter4, HAPPM is introduced, based on the PPA and the ALM. Itproves detailedly that the global convergence of HAPPM exists under certain conditions.In finally, three numerical examples are given to illustrate the efectiveness of the presentalgorithm by compared with the ALM. The last chapter summarizes the whole paper andproposes some research plans for the future.