Filled Function Method And Penalty Function Method For Nonlinear Mixed-integer Programming Problem

Nonlinear mixed-integer programming problem is an important branch of optimization and its application, especially in the filed of engineering which has a lot of models with discrete variables. So solving nonlinear mixed-integer programming problems becomes an important part of mathematic programming. For nonlinear mixed-integer programming, since some variables in feasible solution are discrete number or integer, the classical continuously nonlinear programming methods are no longer useful. Thus it is necessary and meaningful to explore some new methods for nonlinear mixed-integer programming problems.The basic idea of filled function method is to construct a auxiliary functionnamed filled function in the current local minimum.A filled function should hasthe following properties, firstly, the current local minimum should be a local maximum of the filled function;secondly,one can get a better point of objective function though locally searching the filled function. Additionally, the main idea of penalty function method is transform a constrained nonlinear problem into a unconstrained problem and obtain a solution of constrained nonlinear problem though solving many penalty function problem.In this paper,my main interest are firstly,based on the former idea of globally convex filled function method for general nonlinear programming problem, generalizing a definition of globally convex filled function for nonlinear mixed-integer programming and then we present a special formulation of globally convex filled function. The properties and algorithm of this function are discussed too.Secondly, for mixed-integer programming problem with constrains, we make use of the good properties penalty function and transform the problem into unconstrained problems. And furthermore, we prove the equivalency between the global solution of constrained problem and the global solution of the correspondence exactly penalty function.Finally,we serialize the unconstrained problem, i.e.,we transform the original constrained nonlinear mixed-integer programming into a continuous and easy unconstrained nonlinear programming.Chapters are arranged as follows. In chapter one, we introduce the nonlinearmixed-integer programming problem and the current study in and aboard. In chapter two, we study unconstrained mixed-integer problem, which include firstly, preliminaries of filled function; secondly, presenting some necessary definitions anda mixed descent method for local solution of nonlinear mixed-integer programming problems; thirdly, we present globally convex filled function and discuss its properties; and fourthly, we set some example. In chapter three, we study constrained nonlinear mixed-integer programming problems, which include firstly, pre-knowledge of penalty function; secondly, we present a globally exact penalty function for nonlinear mixed-integer programming problem and argue its properties ; serialize the obtained penalty function and prove the equivalency between the solution penalty function and the original problem; fourthly, we consider test examples.In chapter four, sum the paper and present the future work.