Global Optimization Algorithm Of The Auxiliary Function Type

We studied the global optimization method of two kinds problems. One is themethod of the general global optimization problem, that is: a Filled Function methodbased on α-dense for Global Optimization. The other is a specialization problem of thefront kind problem, that is global Optimization Algorithm for Homogeneous PolynomialFunction with Spherical Constraints. They are summarized into two classes: a reducingtransformation and a new filled function. The reducing transformation converts the mul-tivariate global optimization problem into an univariate one, and the filled function basedon the univariate global optimization problem is taken as auxiliary function in order tofind a better minimizer. The paper proposes some kinds of improvement and innovation,seeking to deeply development.At the guidance of the above, the paper is divides into three chapters.Chapter1, we give the background of the global optimization, the knowledge andbasic concept and several local optimization methods are introduced, such as BacktrackingLine Search, steepest descent method, Newton method, quasi-Newton method, conjugategradient method, FR conjugate gradient method, the integral level set method, the tun-nelling method, the filled function method, branch and bound method and so on. Everyalgorithm has its advantages and disadvantages. We will use these local algorithms to findthe local minima.Chapter2, a new method for the global optimization is proposed in this chapter,the method involves structure a new filled function and a reducing transformation. Thereducing transformation converts the multivariate global optimization problem into an uni-variate one, and the filled function based on the univariate global optimization problem istaken as auxiliary function in order to find a better minimizer. For the method, conver-gence to a global minimizer is discussed under some conditions. Some typical examplesare tested to illustrate the efciency of the algorithm.Chapter3, we study the optimization for a generic multi-variate homogeneous poly-nomial function with spherical constraints which is a specialization problem of the previouschapter. First, we convert the constraint problem to unconstraint problem by using polarcoordinate transformation, then use the α dense curve to achieve the dimension reductionefect. Finally, we use the filled function method to solve the lower dimensional uncon-strained problem. At the end of the chapter, algorithm process and the implementation of the algorithm on several test problems are reported, illustrating the efectiveness of thealgorithm studied.