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We suggest a modification of the descent splitting methods for decomposable composite optimization problems, which maintains the basic convergence properties, but enables one to reduce the computational expenses per iteration and to provide computations in a distributed manner. It consists of making coordinate-wise steps together with a special threshold control.

We present an algorithm to locally minimize nonsmooth, nonconvex functions. In order to find descent directions, the notion of quasisecants, introduced in this paper, is applied. We prove that the algorithm converges to Clarke stationary points. Numerical results are presented demonstrating the applicability of the proposed algorithm to a wide variety of nonsmooth, nonconvex optimization problems. We also compare the proposed algorithm with the bundle method using numerical results.

A globally convergent algorithm for equilibrium problems with differentiable bifunctions is proposed. The algorithm is based on descent directions of a suitable family of gap functions. The novelty of the approach is that assumptions which guarantee that the stationary points of the gap functions are global optima are not required.

This article presents a descent method for solving monotone variational inequalities with separate structures. The descent direction is derived from the well-known alternating directions method. The optimal step size along the descent direction also improves the efficiency of the new method. Some numerical results demonstrate that the new method is effective in practice.

In this paper, we consider the optimization method for monotone variational inequality probleln on polyhedral sets. First, we consider the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed and s desirable properties of the merit function are obtained. Under certain assumptions, we show tthat any stationary point of the merit function is a solution of the original problem. A descent metlpod for the optimization problem is proposed and the global convergence of the method is shown

We consider the application of the ABS algorithms, introduced by [I], mainly for the solution of linear algebraic systems, to constrained optimization problems with equality constraints. A general formulation of feasible direction algorithms is obtained and the ABS formulation is given of the reduced gradient method of Wolfe (1967) and of the projected gradient method of Rosen (1960)