Research On A Class Of Precision Penalty Function Algorithms

Penalty function method and sequential quadratic programming method are commonly used to solve nonlinear constrained optimization problems,which transform constrained optimization problems into unconstrained optimization problems.Among them,the penalty function method has been the main research method of experts at home and abroad.If a penalty function of a constrained optimization problem is an exact penalty function,then the minimum point of the penalty problem is the minimum point of the original constrained optimization problem when the penalty parameter is sufficiently large.At present,most of the exact penalty functions studied are simple and non-smooth,so the smoothing of exact penalty functions has become an important research content.The main contents of this paper are as follows:In the first chapter,we mainly introduce the basic knowledge of constrained optimization problems,the research status of penalty function method,the latest research progress of exact penalty function method and the main arrangement of this paper.The second chapter is about the non-differentiability of the exact penalty function of l₁.In this chapter,a smooth approximation of the exact penalty function of l₁ is given,which satisfies:（1）the objective function satisfies the mandatory condition,（2）the optimal solution set of the original inequality constrained optimization problem is a non-empty finite set,（3）the original inequality constrained optimization problem satisfies the KKT second order sufficient condition at any optimal solution set.Under the assumptions of these three conditions,it is proved that if there is at least one optimal solution of the original problem in the strict interior of the feasible domain,then when the penalty parameter is large enough,the optimal solution of any smooth penalty problem must be the optimal solution of the original problem.Based on this penalty function,an algorithm is designed,the convergence of the new algorithm is proved,and the effectiveness of the algorithm is illustrated by a numerical example.In chapter 3,the smooth function in chapter 2 is further improved,and a new penalty function is obtained.under the assumption of chapter 2,its exact penalty is proved.finally,the algorithm is given and the convergence of the new algorithm is proved.a numerical example is given to illustrate the effectiveness of the algorithm.The fourth chapter makes a further summary of the research content of this paper,and looks forward to the future research direction.