Variable Metric Method for Minimization

Abstract

This is a method for niimerically determining local minima of differentiable functions of several variables. In the process of locating each minimum, a matrix is determined which characterizes the behavior of the function about the minimum. For a region in which the function depends quadratically on the variables, no more than N iterations are required, where N is the nunnber of variables. By suitable choice of starting values and without modification of the procedure, linear constraints can be imposed upon the variables.

An extension of davidon methods to non differentiable problems

“…In these methods, a sequence {xn} is generated according to xn + 1 = x~ + p,s~ where p, ~ R is the step size, and sn E H the direction. Adachi [1] points out that many Davidon methods (at least 12) are equivalent in the quadratic case, in the sense that they generate sequences of directions which are colinear, Zoutendijk [1 (3] remarks that, if g, = f'(xn) is the sequence of gradients, then, for each n, all the gradients have the same projection onto s~: (s~, gi) is a constant with respect to i. McCormick and Ritter [7] derive an algorithm in connection with this remark.…”

Forced convective mass transfer from circular disks

“…The programme was devised to determine the minimum of a function of several variables by the use of an iteratire procedure in accordance with the method of Davidon [32]. The following two correlations were obtained.…”

Conjugate gradient and steepest descent approach on quasi-Newton search direction

“…Quasi-Newton method is the well known method to solve the unconstrained optimization problems x f n x min (1) where f is twice continuously differentiable function from n into . Quasi-Newton method is an iterative method, whereby at the (k+1) th iteration, 1 k x is given by (2) where k d denotes the search direction and k is its step length.…”

“…The DFP method was first introduced by [1] (also can be found in [2] and developed by [3]). Hence it was named as DFP (Davidon, Fletcher and Powell) method.…”