Smoothing And Decomposition Method Of Penalty Function

Constrained nonlinear programming problems abound in many important fieldssuch as engineering, national defence, finance etc. One of the main approaches forsolving constrained nonlinear programming problems is to transform it into un-constrained nonlinear programming problem, which solved by using unconstrainedoptimization method. Penalty function method is one of the prevailing approachesto implement the transformation. Penalty function methods seek to obtain the solu-tions of constrained programming problem by solving one or more penalty problems.In the late sixties of the last century, firstly Eremin and Zangwill proposed the con-cept of exact penalty function, from then on, the research of exact penalty functionattracted many scholars'attention. In recent years, some scholars o?er the lowerorder penalty function and discuss its some fundamental properties, and get somevery good theoretical results. Because the lower order penalty function is not dif-ferential, it is hard to compute it in the practical examples. In this thesis, one ofresearch is the smoothing of the lower order penalty function of the general formand smoothed penalty function's properties.In real life, decomposition methods can apply to many fields, e.g. Multi-regionalpower system analysis,Network design,Price-and-resource directive schemes, andso on. Early in the sixties, some decomposition methods were proposed, such asDantzig-Wolfe decomposition and Bender decomposition, Later in the eighties andnineties, a number of relevant research results about that came out. The aim ofdecomposition in optimization is to substitute a large-scale optimization problemsthat present a structure of interrelated subsystems by solutions of subproblems. Inthis thesis, the other research is decomposition methods in the penalty problemwith quadratic penalty term and the application of nonlinear Gauss-seidel methodin constrained optimization problem.The present thesis is organized as follows. In Chapter 1, we give a brief intro-duction to the existing research work on exact penalty functions and decompositionmethods in Constrained programming problems. In Chapter 2, we review the local or global exact penalty property of lower order exact penalty function. Since thelower order exact penalty function is not di?erential, generally speaking, gradient-based optimization methods can not be used directly to solve the lower order penaltyproblem. To avoid this drawback, we propose a smoothing approximation to thelower order exact penalty function of the general form, where we o?er a piecewisefunction with single parameter to smooth approximately. It is shown that whenq is su?ciently large and the global minima of the smoothed lower order penaltyproblem is an -feasible solution to the primal problem for a given , the globalminima of smoothed lower order penalty problem is also the approximate globalminima of the primal problem. The following numerical examples show that it isapplicable to solve the smoothed lower order penalty problem in order to solve theprimal problem.From the Chapter 1 or 2, we know that an approximate global minimizer ofoptimization problem with constrains can be obtained by solving its correspondingunconstrained penalty problem, but when penalty problem is relatively large-scaleoptimization problems, here we may use decomposition method to decompose it.In Chapter 3, for separable problem with one equality constrain, we give its un-constrained penalty problem with quadratic penalty term, o?er the decompositionmethods to penalty problem and propose the corresponding algorithm. we alsoapply nonlinear Gauss-seidel decomposition techniques to the constrained optimiza-tion problem and o?er the corresponding algorithms . Some numerical examplesare given to illustrate the applicability of the presented decomposition methods bycomparing them with the auxiliary problem principle(APP) method and the blockcoordinate descent (BCD) method, which are decomposition methods to the aug-mented Lagrangian function. In Chapter 4, we conclude the thesis and Look forwardto the future .