method of dual matrices
Springer Online Journal Archives 1860-2000
Abstract 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 mostn + 1 iterations, wheren 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 mostn + 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.
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