Local convergence of the steepest descent method in Hilbert spaces

Nonsmooth Steepest Descent Method by Proximal Subdifferentials in Hilbert Spaces

In this paper, we first study a nonsmooth steepest descent method for nonsmooth functions defined on a Hilbert space and establish the corresponding algorithm by proximal subgradients. Then, we use this algorithm to find stationary points for those functions satisfying prox-regularity and Lipschitz continuity. As an application, the established algorithm is used to search for the minimizer of a...

A Steepest Descent Method for Oscillatory Riemann-hilbert Problems

but it will be clear immediately to the reader with some experience in the field, that the method extends naturally and easily to the general class of wave equations solvable by the inverse scattering method, such as the KdV, nonlinear Schrödinger (NLS), and Boussinesq equations, etc., and also to "integrable" ordinary differential equations such as the Painlevé transcendents. As described, for...

An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces

Mann-type Steepest-descent and Modified Hybrid Steepest-descent Methods for Variational Inequalities in Banach Spaces

1Department of Mathematics, Shanghai Normal University, Shanghai; and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, China 2Department of Mathematics & Statistics, College of Science, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia; and Department of Mathematics, Aligarh Muslim University, Aligarh, India 3Department of Applied Mathematics, National S...

On the Steepest Descent Method for Matrix

We consider the special case of the restarted Arnoldi method for approximating the product of a function of a Hermitian matrix with a vector which results when the restart length is set to one. When applied to the solution of a linear system of equations, this approach coincides with the method of steepest descent. We show that the method is equivalent with an interpolation process in which the...