Hybrid Steepest Descent Method Research Articles

STRONG CONVERGENCE THEOREMS OF RELAXED HYBRID STEEPEST-DESCENT METHODS FOR VARIATIONAL INEQUALITIES

Assume that F is a nonlinear operator on a real Hilbert space H which is η-strongly monotone and κ-Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed point sets of a finite number of nonexpansive mappings on H. We develop a relaxed hybrid steepest-descent method which generates an iterative sequence {xn} from an arbitrary initial point x0 ∈ H. The sequence {xn} is shown to converge in norm to the unique solution u∗ of the variational inequality F(u ∗), v − u ∗ ≥ 0 ∀v ∈ C under the conditions which are more general than those in Ref. 19. Applications to constrained generalized pseudoinverse are included.

STRONG CONVERGENCE THEOREMS OF RELAXED HYBRID STEEPEST-DESCENT METHODS FOR VARIATIONAL INEQUALITIES

Assume that $F$ is a nonlinear operator on a real Hilbert space $H$ which is $\eta$-strongly monotone and $\kappa$-Lipschitzian on a nonempty closed convex subset $C$ of $H$. Assume also that $C$ is the intersection of the fixed point sets of a finite number of nonexpansive mappings on $H$. We develop a relaxed hybrid steepest-descent method which generates an iterative sequence $\{x_n\}$ from an arbitrary initial point $x_0 \in H$. The sequence $\{x_n\}$ is shown to converge in norm to the unique solution $u^*$ of the variational inequality \[ \langle F(u^*), v-u^* \rangle \geq 0 \quad \forall v \in C \] under the conditions which are more general than those in Ref. 19. Applications to constrained generalized pseudoinverse are included.

Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities

Assume that F is a nonlinear operator on a real Hilbert space H which is η-strongly monotone and κ-Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed point sets of a finite number of nonexpansive mappings on H. We devise an iterative algorithm which generates a sequence (xn) from an arbitrary initial point x0∈H. The sequence (xn) is shown to converge in norm to the unique solution u* of the variational inequality $$\left\langle {F(u*),\user1{v} - u*} \right\rangle \geqslant 0$$ Applications to constrained pseudoinverse are included.

Computation of symmetric positive definite Toeplitz matrices by the hybrid steepest descent method

This paper studies the problem of finding the nearest symmetric positive definite Toeplitz matrix to a given symmetric one. Additional design constraints, which are also formed as closed convex sets in the real Hilbert space of all symmetric matrices, are imposed on the desired matrix. An algorithmic solution to the problem given by the hybrid steepest descent method is established also in the case of inconsistent design constraints.

Nonstrictly Convex Minimization over the Bounded Fixed Point Set of a Nonexpansive Mapping

In this paper, we consider, in a finite dimensional real Hilbert space , the variational inequality problem VIP: find , where is nonexpansive mapping with bounded and is paramonotone and Lipschitzian over . The nonstrictly convex minimization over the bounded fixed point set of a nonexpansive mapping is a typical example of such a variational inequality problem. We show that the hybrid steepest descent method, of which convergence properties were examined in some cases for example (Yamada, I. (2000). Convex projection algorithm from POCS to Hybrid steepest descent method. The Journal of the IEICE (in Japanese) 83:616–623; Yamada, I. (2001). The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S., eds. Inherently Parallel Algorithm for Feasibility and Optimization. Elsevier; Ogura, N., Yamada, I. (2002). Non-strictly convex minimization over the fixed point set of an asymptotically shrinking nonexpansive mapping. Numer. Funct. Anal. Optim. 23:113–137), is still applicable to the case where and T satisfy the above conditions.

NON-STRICTLY CONVEX MINIMIZATION OVER THE FIXED POINT SET OF AN ASYMPTOTICALLY SHRINKING NONEXPANSIVE MAPPING

ABSTRACT Suppose that T is a nonexpansive mapping on a real Hilbert space satisfying for some R > 0. Suppose also that a mapping is κ-Lipschitzian over and paramonotone over . Then it is shown that a variation of the hybrid steepest descent method (Yamada, Ogura, Yamashita and Sakaniwa (1998), Deutsch and Yamada (1998) and Yamada (1999–2001)): generates a sequence (un ) satisfying , when is finite dimensional, where for all is the solution set of the variational inequality problem . This result relaxes the condition on and (λ n ) of the hybrid steepest descent method (Yamada (2001)), and makes the method applicable to the significantly wider class of convexly constrained inverse problems as well as the non-strictly convex minimization over the fixed point set of asymptotically shrinking nonexpansive mapping.