Research On The Smoothing Of The Exact Penalty Function For Constrained Optimization Problems

Optimization theory and methods include unconstrained optimization theory,con-strained optimization theory,non-smooth optimization theory and calculation method,which is widely used in economic planning,engineering design,production management,trans-portation and other areas.The constrained nonlinear programming problem is the most common optimization problem.The constrained nonlinear programming problem can be transformed into an unconstrained nonlinear programming problem under some conditions.The penalty function method is to get the solution of the constrained programming problem by solving the unconstrained penalty problem.The exact penalty function method is a kind of classical methods to solve the optimization problem.The exact penalty function is that the minimum of the penalty problem is the minimum of the original problem or the mini-mum of the original problem is the minimum of the penalty problem when the parameter is large enough.Most of the exact penalty functions studied by scholars recently are simple and non-smooth.The simple penalty function is that the gradient information of the objective function and the constraint function does not contain in the penalty function.The study on smoothness of the exact penalty function has received extensive attention.The main work of this article is to study the smoothness of the exact penalty function.There are four chapters in this thesis:In chapter 1,the author gives a brief description of constrained optimization problems and related knowledge,penalty function methods,and the main,work of this paper.In chapter 2,a new smoothing method for low order exact penalty functions is proposed.And it is proved that the optimal solution of the smooth penalty problem is the approximate optimal solution of the original problem.At the same time,based on the smooth penalty function,an algorithm is proposed and proved that the algorithm is global convergent under weak conditions.Numerical examples are given to illustrate the efficiency of the proposed algorithm.In chapter 3,the author proposes a new smoothing method for the l1 exact penalty function.Under some conditions,an approximate global solution of the the original problem can be obtained by searching a global solution of the smooth penalty problem.Based on the smoothed penalty function,the author develops an algorithm.Meanwhile,the algorithm is shown to be global convergent under some mild conditions.The efficiency of the algorithm is illustrate with some numerical examples.In chapter 4,the author summarizes the specific work and proposes the problems in this paper and prospect for future study.