Research On Exact Penalty Function Of Constrained Optimization Problems

In real life,we often meet the problems which select an optimal strategy among many strategies.For example,how we choose the material collocation ratio to make the quality be satisfied and the cost be lowest in the product design.The nonlinear constrained optimization problems are common.A general method of solving these problems is the penalty function method.By the penalty function,constrained nonlinear programming problem can be transformed into unconstrained nonlinear programming problem.The traditional definitional simple penalty functions must not be smooth or exact.The simple penalty function means that the objective function and constrained functions include in the penalty function and their gradients don't include in the penalty function.In this paper,the author constructs the simple smooth exact penalty function.In addition,the lower order exact penalty function is not differentiable,which is difficult to calculate the exact penalty problem.In order to solve this problem,a smooth approximation to the non-differentiable exact penalty function is given.This paper mainly consists of four chapters.In Chapter 1,the author introduces the basic knowledge of constrained optimization problems,the exact penalty function method and the main work of the paper.In Chapter 2,a new class of simple smoothing penalty function is proposed and proved that it is exact.The penalty function algorithm is given and numerical examples show that the proposed algorithm is feasible.In Chapter 3,another new class of simple smooth penalty function is proposed and proved that it is exact.the order of can improve the speed of convergence.The penalty function algorithm is given and numerical examples show that the proposed algorithm is feasible.In Chapter 4,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.