The Penalty Function And Filled Function Methods In Nonlinear Programming

Constrained nonlinear programming problems abound in many important fields such as engineering, national defence, finance etc. One of the main approaches for solving constrained nonlinear programming problems is to transform it into unconstrained nonlinear programming problem. Penalty function methods and Lagrangian duality methods are the two prevailing approaches to implement the transformation. Penalty function methods seek to obtain the solutions of constrained programming problem by solving one or more penalty problems. If each minimum of the penalty problem is a minimum of the primal constrained programming problem, then the corresponding penalty function is called exact penalty function. In this thesis, we first give some penalty function, and then we discuss the global exact and approximatively exact penalty property of exact penalty functions, we also discuss smoothing of exact penalty functionsGlobal optimization problems abound in economic modelling, finance, networks and transportation, databases, chip design, image processing, chemical engineering design and control, molecular biology, and environmental engineering. Since there exist multiple local optima that differ from the global solution, and the traditional minimization techniques for nonlinear programming are devised for obtaining local optimal solution, how to obtain the globally optimal solutions is very important topic. In this thesis, we also discuss the filled function methods for global optimization and give a new filled function.This paper mainly consists of five chapters.In the first chapter, we give a brief introduction to the existing research work on penalty functions.In the second chapter, we give an multiplier penalty function and discuss its properties. Based on the penalty function, an algorithm is given.In chapter three, a kind of smoothing and approximatively exact penalty functions is given, and its approximatively exact property is proved. Finally an algorithm is given.In chapter four, the global exact penalty function is given, we first prove its properties, then an algorithm is given.In the last chapter, for global optimization problems, we give a new algorithm called filled modified tunnelling function methods, an auxiliary function called filled modified tunnelling function is first given, it has good properties of filled function and tunnelling function. Then, based on the function, an algorithm is given. The implementation of the algorithms on several test problems is reported with satisfactory numerical results.