Study On Smoothness And Algorithm Of Exact Penalty Function

Optimization theory and methods became an independent discipline since the simplex algorithm was proposed to solve the linear programming problem in the 40’s of the last century by Dantzig. With the rapid development of computer technology, optimization theory and methods widely in economic, engineering, military and other fields. The constrained nonlinear programming problem is often used. Constrained nonlinear programming problem can often be transformed into unconstrained nonlinear programming problem. Penalty function method is one of the most commonly used methods. It gets the solution of the constrained programming problem by solving the unconstrained penalty problem. Exact penalty function is that when the penalty parameter is sufficiently large, the minimum of the penalty problem is the minimum of the original constraint programming problem. Simple penalty function is that the penalty function contains the constraint functions and the objective function of the original problem and does not contain their gradient information, otherwise,the penalty function is complex. For the traditional penalty function, if it is simple it can not be both exact and smooth. The exact penalty function of the present study is mostly simple and not smooth. In order to apply the gradient of algorithm unconstrained optimization,the smoothing of the exact penalty function becomes particularly important.This paper mainly consists of four chapters.In chapter 1, the author introduces the basic knowledge of constrained optimization problems and the exact penalty function method.In chapter 2, a new smoothing method is proposed for the lower exact penalty function.It is proved that the approximate optimal solution of the smooth penalty problem is the approximate optimal solution of the original problem. An algorithm is designed based on this penalty function. The algorithm can be shown to be convergent under some mild conditions. The efficiency of the algorithm is illustrated with some numerical examples.In chapter 3, the smoothness of the square-root exact penalty function is studied, and a new smoothing method is presented based on the smooth penalty function. It is proved that the approximate optimal solution of the smooth penalty problem is the approximate optimal solution of the original problem, and it is proved that the algorithm is convergent.Finally, some numerical examples are illustrated to show that the algorithm is efficient.In chapter 4, the author propose a method to smooth l1 exact penalty function for inequality constrained optimization. It is shown that an approximate solution of the original problem can be obtained by searching a solution of the smoothed penalty problem. Under some mild conditions, the method based on the smoothing function is shown to be globally convergent. Some numerical examples are given to illustrate the applicability of the present smoothing method.