Research On The Smoothing Of Objective Penalty Function Method For Inequality Constrained Optimization Problems

Optimization theory and methods is based on the simplex algorithm which was proposed to solve linear programming problem by Dantzig in 1947.With the rapid development of computer technology,optimization theory and methods became an independent discipline.Optimization theory and methods is widely used in science and technology,economic,military and other fields.The most widely used is constrained nonlinear programming problem.Constrained nonlinear programming problem can often be transformed into unconstrained nonlinear programming problem.One of important methods for solving nonlinear programming problems is the penalty function method.The penalty function method gets the solution of the constrained programming solution by solving the unconstrained penalty problem.Exact penalty function is that the minimum of the penalty problem is the minimum of the original constraint programming problem or the minimum of the original constraint programming problem is the minimum of the penalty problem when the penalty parameter is sufficiently large.For the traditional penalty function,if it is simple and smooth,it can not be exact;if it is simple and exact,it can not be smooth.The main job of this paper is to study the objective penalty functions which are different with the traditional penalty functions,and propose new objective penalty functions and discuss their smoothing.The structure of this paper is organized as follows:In Chapter 1,the author introduces the basic concept and knowledge of the optimization problems and the objective penalty functions,and then the main job of the paper is introduced.In Chapter 2,for nonlinear constrained optimization problems,the author studies a new objective penalty function,proves its accuracy,gives smoothing approximate to objective penalty function and defines the corresponding smoothed objective penalty problem.Error estimations among the optimal objective values of the objective penalty problem and the smoothed objective penalty problem are obtained.Based on the smoothed objective penalty function,the author develops an algorithm and proves the global convergence of the algorithm.Numerical experiments are provided to show that the proposed algorithm is efficient.In Chapter 3,another new exact penalty function for nonlinear constrained optimization problems,The author studies its smoothing and defines the corresponding objective penalty optimization problem.Error estimations among the optimal objective values of the objective penalty problem and the smoothed objective penalty problem are obtained.An algorithm based on the smoothed objective penalty function is proposed and proved its global convergence.Numerical experiments are provided to show the efficiency of the proposed method.In Chapter 4,the research contents of this paper are summarized,and the directions for further research are proposed.