Filled Function Methods For Nonsmooth Global Optimization

Frequently, practitioners need to solve global optimization problems in many fields such as engineering design, financial management, bioengineering, and social science. So global optimization becomes a crucial computational task for researchers, which discusses the characters of global optimal choice of multivariate nonlinear functions on a constrained region and constructs computing approaches to find the global optimal solution, as well as discusses the theoretical properties and calculation properties of the solutions. However, due to the existence of multiple local minimiz-ers that differ from the global solution, we have to face two difficulties: how to jump from a local minimizer to a smaller one and how to judge that the current minimizer is a global one. Hence all these problems cannot be solved by classical nonlinear programming techniques directly. Generally speaking, the global optimization methods can be divided into two types: stochastic methods and deterministic methods. The filled function method, first proposed for smooth optimization by Ge and Qin (1987), is one of the effective deterministic global optimization methods for settling the first difficulty. It modifies the objective function as a filled function, and then finds a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. The filled function method provides us with a good idea to use the local optimization techniques to solve global optimization problems.Nonsmooth optimization is also one of the active research areas in computational mathematics, applied mathematics, and engineering design optimization. Most of the existing nonsmooth optimization methods are actually to seek a local minimizer. Several researchers have extended the filled function method from smooth global optimization to nonsmooth global optimization. The existing filled functions have some drawbacks such as requiring that the objective function has only a finite number of local minimizers, or the parameters of filled functions heavily restricted by the minimal basin radius of local minimizers, or requiring that the filled function has a minimizer on the line. All of these characteristics are strongly undesirable in numerical applications as they are liable to the illness of computation. Therefore, further research is worthy of continuing on how we can construct filled functions with simple forms, better properties and more efficient algorithms.Based on the current status of research scholars, for a number of outstanding issues, the aim of this paper is to develop the filled function with certain satisfactory properties in nonsmooth global optimization. This paper mainly consists of five chapters. In Chapter 1, we give a brief introduction to the existing research work on global optimization and nonsmooth optimization. These methods include the filled function method, the tunnelling method, the penalty function method, and so on, with an emphasis on the filled function method in smooth optimization.In Chapter 2, we discuss the filled function to find a global minimizer for a general class of nonsmooth programming problems with a closed bounded domain. Based on the new definition for the filled function, we propose a two-parameter filled function to improve the efficiency of numerical computation. This filled function needs not the assumption that the objective function is diflFerentiable and has a finite number of local minimizers, moreover, its parameters are easy to set. Owing to the complicated co-adjustment of the two-parameters, attempts have been made to improve the properties of the filled function. The basic idea for the modification is to cancel a parameter. Then we propose a new one-parameter filled function to eliminate these drawbacks. Based on these analyses, two corresponding filled function algorithms are presented. Numerical results obtained indicate the efficiency and reliability of the proposed filled function methods. The performance of the proposed filled function methods is compared to the performance of some well-known global optimization methods, the interval method and the tunneling method. For the questions on how to evaluate the convergence and how to decide a search direction in finding another better local minimizer, we give the corresponding suggestions.In Chapter 3, we extend the idea for nonsmooth unconstrained global optimization to nonsmooth inequalities constrained global optimization. Under the def- inition for the filled function in nonsmooth constrained optimization, we give a two-parameter filled function and a one-parameter filled function, and present two corresponding filled function algorithms. The implementation of algorithms on several test problems is reported with satisfactory numerical results.In Chapter 4, we extend the idea for nonsmooth unconstrained global optimization to nonsmooth equalities constrained global optimization. We proposes a two-parameter filled function and the corresponding filled function algorithm.In Chapter 5, extension conceivable applications are given in order to evaluate the merits of the filled function method. The idea of finding a global minimizer by using filled function can be explored in a number of fields such as multiobjective programming, license plate recognition system, face recognition system, granular computing. Prom this point of view, a lot of operations can be defined and relations among them can be studied. This opens an extensive area for research and, hopingly, puts forward an interesting way for utilization of global optimization to modeling of phenomena.