Transformation Function Methods For Nonlinearly Global Optimization

The optimization is a widely used discipline, which discusses the characters of optimal choice on decision problems and constructs computing approaches to find the optimal solution. Due to the advancement of society and the development of science and technology, the optimization problems are often discovered in the field of economic planning administration, engineering design, production management, traffic transportation, national defence and so on. They are so important that meet with much recognition. With the speedy development of computer and the hard work of scientists, the theoretic analysis and computational methods on optimization have been highly improved .This paper mainly consists of five chapters.In the first chapter, some mainly methods for global optimization problems are briefly presented. And several basis concepts and characters on generally nonlinear programming are introduced.In the second and third chapter the transformation functions for unconstrained global optimal problems are mainly discussed. To find the effective methods for finding the global optimal solutions of a general multi-mininiizers function is one of the hot topics. There two difficulties in global optimization. One is how to leave from a local minimizer to a smaller one and the other is how to judge that the current minimizer is global. The tunnelling function proposed by Levy and Montalvo (1985) and filled function algorithms introduced by Ge and Qin (1987) are two well-known and practical methods for settling the first difficulty. They have common character. If a local minimizer x*1 has been found, we can make a auxiliary function, such as tunnelling function or filled function, such that iterative sequential points leave the valley in which x*1 lies to find a better point x' in the lower valley (i.e. f(x') < f(x*1). Then let x' be a new initial point to search for a better minimizer. In second chapter two classes of transformation functions for global optimization are defined and it is proved theoretically and computationally that they possess the both characters of tunnelling functions and filled functions under some general assumptions. In third chapter some easy and computable transformation functions are presented. They have the both characters of tunnelling functions and filled functions as well. We proved the main characters of transformation functions, that is, the transformation functions have no any minimizer or stationary point on the region {x : f(x) ≥ f(x*1)} and have at least one minimizer on the region {x : f(x) < f(x*1)} if {x : f(x) ≥ f(x*1)} ≠ 0. Certainly, the numerical results are listed in these two chapters.In chapter four, the idea for unconstrained global optimization is extended to nonlinear global problems with constraints. First, there exist many effective methods for local minimizers of nonlinear programming. Because these methods...