The Penalty Function Of Equality Constraint Optimization And Minimax Problem

In real life,we always encounter in numerous optimization problems to select the type of makes efficient use of resources or the biggest target cost minimum.According to the constrains,it is divided into constrained optimization problems and unconstrained optimization problems.These two kinds of problems are quiet different in the aspect of theoretical analysis and algorithm design,in some cases,these can be converted to each other.In general,unconstrained optimization problems are easier to be solved than constrained optimization problems.In this paper,choosing the penalty function method can solve the constrained optimization problems.In other words,using the penalty functions can convert the constrained problems to unconstrained problems.For the traditional penalty function,if it is simple and smooth,it must not be exact;if it is simple and exact,it must not be smooth.So the main job of this thesis is to modify the traditional penalty functions,and make the simple penalty function be exact and smooth.The structure of this paper is organized as follows:In Chapter 1,the main content is to introduce the basic concept and knowledge of the optimization problems and the penalty functions,and then the main work of the paper is introduced.In Chapter 2,for the equality constrains optimization problems,a class of penalty functions are derived by adding a variable on constraint functions.Using the K-K-T constrains and Lagrange function,the class of penalty functions is proved to be smooth and exact on the closed interval.An algorithm is given to solve optimization problems with equality constrains.Numerical experiments illustrate that the presented algorithm is effective.In Chapter 3,for the equality constrains optimization problems,a new class of penalty functions are derived.The class of penalty functions is proved to be smooth and exact.And an algorithm is proved to solve optimization problems and some numerical experiments illustrate that the presented algorithm is effective.In Chapter 4,for inequality constrains optimization problems,a new class of objective exact penalty functions is derived by an objective penalty parameter and a constraint penalty parameter which can both penalize the objective function and the constraint functions.The aim is to make the constraint violation measure reduce and the objective function close to the optimum value.Furthermore,two algorithms for the global or local solution are presented and proved that they are convergent.Numerical experiments illustrate that the presented algorithms are effective.