Method of Dual Matrices for Function Minimization

作     者：Huang, H. Y. 

作者机构：Rice Univ Dept Mech & Aerosp Engn & Mat Sci Houston TX 77251 USA 

出 版 物：《JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS》 (优选法理论与应用杂志)

年 卷 期：1974年第13卷第5期

页      面：519-537页

核心收录：

学科分类：1201[管理学-管理科学与工程(可授管理学、工学学位)] 07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：National Science Foundation [GP-32453] 

主　　题：Mathematical programming function minimization method of dual matrices computing methods numerical methods 

摘      要：In this paper, the method of dual matrices for the minimization of functions is introduced. The method, which is developed on the model of a quadratic function, is characterized by two matrices at each iteration. One matrix is such that a linearly independent set of directions can be generated, regardless of the stepsize employed. The other matrix is such that, at the point where the first matrix fails to yield a gradient linearly independent of all the previous gradients, it generates a displacement leading to the minimal point. Thus, the one-dimensional search is bypassed. For a quadratic function, it is proved that the minimal point is obtained in at most n + 1 iterations, where n is the number of variables in the function. Since the one-dimensional search is not needed, the total number of gradient evaluations for convergence is at most n + 2. Three algorithms of the method are presented. A reverse algorithm, which permits the use of only one matrix, is also given. Considerations pertaining to the applications of this method to the minimization of a quadratic function and a nonquadratic function are given. It is believed that, since the one-dimensional search can be bypassed, a considerable amount of computational saving can be achieved.