An Exact Penalty Function Method For Solving Constrained Optimization Problems

For constrained optimization problems with equality,inequality and bound con-straints,we propose a new class of generalized exact penalty functions with smooth,nonsmooth constrained optimization problems respectively,including many com-mon exact penalty functions as special cases.When the objective function and the constraint function are smooth and only the equality constraints are considered,the penalty function is the penalty function in[25],and when the penalty function is taken as a special case,it becomes the penalty function in[13].It is proved ex-act,under some suitable conditions,in the sense that each local optimizer of the penalty function corresponds to a local optimizer of the original problem.Further-more,necessary and sufficient conditions are discussed for the inverse proposition of exactness penalization in nousmooth cases.Based on the these results,a class of penalty algorithms with possible infeasibility is presented,including the global convergence property and numerical experiments.The main contents of this article are as follows.The first chapter is the introduction part.First,we introduce the research background and present situation.Secondly,we intxoduce the significance and main research content of this paper.In Chapter 2,for smooth constrained optimization problems with equality,in-equality and bound constraints,we propose a new class of generalized exact penalty functions.Firstly,under appropriate conditions,we prove that the each local opti-mizer of the penalty function corresponds to a local optimizer of the original prob-lem.Secondly,in the case that the problem may not be feasible,the corresponding penalty function method is proposed.Furthermore,the effectiveness of infeasible detection is proved and the global convergence analysis of the algorithms is given.Finally,the availability of the algorithm is illustrated by numerical experiments.In Chapter 3,for the constrained optimization problems that the objective function and the constraint function may be nonsmooth,we give a class of smooth?ing exact penalty functions.Under satisfying the condition of weak generalized Mangasaxian Fromovitz constraint qualification,we prove the exactness of penalty function in the sense of local optimal solution.Furthermore,necessary and sufficient conditions are discussed for the inverse proposition of exact penalization.Sinilarly,we present a class of smoothing penalty algorithms which could deduce whether the problem is feasible in the finite step.It is worth noting that in the given algorithms,the exact solution of the subproblems is not required,we allow to obtain a solution which has certain error when solving subproblems,that is,it can prove the global convergence of the algorithms.