On The Smoothing Of The Exact Penalty Function

With the rapid development of computer technology, optimization theory and method arewidely used in the national economy, military, science and other fields. Constrained nonlinearprogramming problems abound in many important fields such as finance, national defense,engineering etc. One of the main approaches for solving constrained nonlinear programmingproblems is to convert it into unconstrained nonlinear programming problem. Penalty functionmethod is one of the methods of conversion. Penalty function methods get the solutions ofconstrained programming problem by solving one or more penalty problems. When theparameter is large enough and each minimum of the penalty problem is a minimum of theprimal constrained programming problem, the corresponding penalty function is called exactpenalty function. But current research on most exact penalty function is simple and nonsmooth, these make it difficult to apply fast unconstrained algorithms based on the gradients.In this thesis, the main study is on the smoothing of exact penalty functions.This paper mainly consists of three chapters.In the first chapter, the author gives a brief introduction to penalty functions and theproperties or theorem of optimization problems.In the second chapter, the author proposes a method to smooth the square-root exactpenalty function for inequality constrained optimization in global optimization. Under someconditions, an approximate global solution of the original problem can be obtained bysearching a global solution of the smoothed penalty problem. The author develops analgorithm based on the smoothed penalty function. The algorithm is shown to be globallyconvergent under some mild conditions. Some numerical examples are given to illustrate theapplicability of the given smoothing method.In the third chapter, the author proposes a method to smooth the general lower orderexact penalty function for inequality constrained optimization in global optimization. Undersome conditions, an approximate global solution of the original problem can be obtained bysearching a global solution of the smoothed penalty problem. The author develops analgorithm based on the smoothed penalty function. The algorithm is shown to be globallyconvergent under some mild conditions. Some numerical examples are given to illustrate theapplicability of the given smoothing method.