Naokazu Yamaki

A BOUNDED DETERIORATION PROPERTY OF A SYMMETRIC POSITIVE DEFINITE CLASS OF NEWTON-LIKE METHODS AND ITS APPLICATION

This paper is concerned with Newton-like methods for solving unconstrained minimization problems. We derive a general form and its factorized form of a symmetric positive definite matrix that satisfies the secant condition in order to approximate the Hessian matrix of the objective function. We obtain the bounded deterioration property for such a general form, which is an extention of the bounded deterioration property for secant methods. Applying the general form to the secant method, we obtain a new family that includes the Broyden family, and we show local and q-superlinear convergence of our method. Furthermore, we propose applying the general form to nonlinear least squares problems to obtain a modification of the Gauss-Newton method.

A model-based AHP Including the Case of Incomplete Information

Sekitani and Yamaki introduced two new concepts of self-evaluation value and non-self-evaluation value into AHP and showed that the eigenvalue method can be formulated as some mathematical programming problems with the ratios of the self-evaluation value to the non-self-evaluation value. This study develops a new discrepancy-minimization problem with the ratios of the self-evaluation value to the non-self-evaluation value and their reciprocals. We show that its optimal solution is identical to the principal eigenvector of the pairwise comparison matrix. We compare the analytical properties of the proposed optimization model with that of Harker method for the case of AHP with incomplete information.