Smoothing Of Penalty Function

Constrained nonlinear programming problems abound in many important fields such as engineering,economy,national defence etc.One of the main methods for solv-ing constrained nonlinear programming problems is to transform it into uncon-strained nonlinear programming problem,which solved by using unconstrained op-timization method.Penalty function method is one of the prevailing approaches to implement the transformation.Penalty function methods obtain the solutions of con-strained programming problem by solving some penalty problems.In twenty century and sixties,Zangwill and Eremin first proposed the concept of exact penalty func-tion,from then on,many scholars pay attention to the research of exact penalty func-tion.In recent years,some scholars offer the lower order penalty function and discuss its some fundamental properties,and get some very good theoretical results.Because the lower order penalty function is not differential,the compute is hard.In this the-sis,one of research is the smoothing of the lower order penalty function of the general form and smoothed penalty function's properties. In this thesis,the other research is using exponential function to smooth e∞exact penalty function,and propose some pertinent characters.The present thesis is organized as follows.In Chapter l,we give a introduction to the existing research work on exact penalty functions and the smoothing method of minimax problem.In Chapter 2,we review the local or global exact property of lower order exact penalty function.Since the lower order exact penalty function is not differential,generally speaking,gradient-based optimization methods can not be used directly to solve the lower order penalty problem.To overcome this difficulty,we propose a smoothing approximation to the lower order exact penalty function of the general form,where offer a piecewise function with pair of parameters to smooth approximately.It is shown that when q is sufficiently large and the global minima of the smoothed lower order penalty problem is anε-feasible solution to the original problem for a givenε,the global minima of smoothing lower order penalty problem is the approximate global minima of the original problem.the numerical examples show that it is applicable to solve the smoothing lower order penalty problem in order to solve the original problem. In Chapter 3,we offer exponential function to smooth e∞exact penalty function,and propose some pertinent characters.In Chapter 4,we conclude the thesis and Look forward to the future.