In this paper, the concept of the s-doubly diagonally dominant matrices is introduced and the properties of these matrices are discussed. With the properties of the s-doubly diagonally dominant matrices and the properties of comparison matrices, some equivalent conditions for H-matrices are presented. These conditions generalize and improve existing results about the equivalent conditions for H-matrices. Applications and examples using these new equivalent conditions are also presented, and a new inclusion region of k-multiple eigenvalues of matrices is obtained. § 1 Introduction Special matrices, especially H-matrices and M-matrices ,have very wide applications in numerical calculations, control theory, mathematical physics, optimization techniques and so on. In recent two or three decades, the studies in these matrices are fruitful, and many graceful equivalent conditions to M-matrices have been proposed. By contrast, though the H-matrices are closely related with M-matrices, researches on H-matrices show that the problems in H-matrices are more difficult and some results of M-matrices may not hold for H-matrices. [4] and E3] give some new sufficient conditions for nonsingularity of matrices using properties of comparison matrices and topology methods, respectively. These conditions improve the results in [8--11]. It is motivated from their proposed methods that the sufficient conditions about H-matrices can be weakened. In this paper, we present some weak equivalent conditions of H-matrices by defining a class of s-doubly diagonally dominant matrices and using the properties of comparison matrices. Properties of the sdoubly diagonally dominant matrices are also discussed. These new equivalent conditions Received : 2004-02-19. MR Subject Classification.. 15A06, 15A57, 65F05.

A new trust region method is presented, which combines nonmonotone line search technique, a self-adaptive update rule for the trust region radius, and the weighting technique for the ratio between the actual reduction and the predicted reduction. Under reasonable assumptions, the global convergence of the method is established for unconstrained nonconvex optimization. Numerical results show that the new method is efficient and robust for solving unconstrained optimization problems.

In this paper, we solve multi-objective programming problems with equilibrium constraints by means of homotopy interior point algorithm. We also prove the global convergence of this homotopy interior point algorithm under assumptions. Moreover, the results of the numeric example shows that this method is feasible and effective.

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