Alexander J. Zaslavski

Asymptotic Behavior of Relatively Nonexpansive Operators in Banach Spaces

Abstract Let K be a closed convex subset of a Banach space X and let F be a nonempty closed convex subset of K. We consider complete metric spaces of self-mappings of K which fix all the points of F and are relatively nonexpansive with respect to a given convex function ƒ on X. We prove (under certain assumptions on ƒ) that the iterates of a generic mapping in these spaces converge strongly to a retraction onto F.

The structure of extremals of a class of second order variational problems

Abstract We study the structure of extremals of a class of second order variational problems without convexity, on intervals in R+. The problems are related to a model in thermodynamics introduced in [7]. We are interested in properties of the extremals which are independent of the length of the interval, for all sufficiently large intervals. As in [12, 13] the study of these properties is based on the relation between the variational problem on bounded, large intervals and a limiting problem on R+. Our investigation employs techniques developed in [10, 12, 13] along with turnpike techniques developed in [16, 17].

Variational Principles and Well-Posedness in Optimization and Calculus of Variations

The concluding result of the paper states that variational problems are generically solvable (and even well-posed in a strong sense) without the convexity and growth conditions always present in individual existence theorems. This and some other generic well-posedness theorems are obtained as realizations of a general variational principle extending the variational principle of Deville--Godefroy--Zizler.

Nonlinear Analysis and Optimization II: Optimization

We prove a necessary optimality condition for isoperimetric problems on time scales in the space of delta-differentiable functions with rd-continuous derivatives. The results are then applied to Sturm-Liouville eigenvalue problems on time scales.

A Sufficient Condition for Exact Penalty in Constrained Optimization

In this paper we use the penalty approach to study three constrained minimization problems. A penalty function is said to have the exact penalty property [J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, 2 vols., Springer-Verlag, Berlin, 1993] if there exists a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish a very simple sufficient condition for the exact penalty property.

Convergence and perturbation resilience of dynamic string-averaging projection methods

We consider the convex feasibility problem (CFP) in Hilbert space and concentrate on the study of string-averaging projection (SAP) methods for the CFP, analyzing their convergence and their perturbation resilience. In the past, SAP methods were formulated with a single predetermined set of strings and a single predetermined set of weights. Here we extend the scope of the family of SAP methods to allow iteration-index-dependent variable strings and weights and term such methods dynamic string-averaging projection (DSAP) methods. The bounded perturbation resilience of DSAP methods is relevant and important for their possible use in the framework of the recently developed superiorization heuristic methodology for constrained minimization problems.

The set of noncontractive mappings is σ-porous in the space of all nonexpansive mappings☆

Abstract In a recent paper we have shown that most nonexpansive mappings (in the sense of Baires categories) are, in fact, contractive. More precisely, we have equipped the space of all nonexpansive mappings with a natural complete metric and proved that a generic element in this space is contractive. In the present paper we show that the set of all noncontractive mappings is not only of the first category, but also σ-porous.

Generic Well-Posedness of Optimal Control Problems without Convexity Assumptions

The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions. In A. J. Zaslavski [Nonlinear Anal., to appear], a generic existence and uniqueness result (with respect to variations of the integrand of the integral functional) without the convexity condition was established for a class of optimal control problems satisfying the Cesari growth condition. In this paper we extend the generic existence and uniqueness result in A. J. Zaslavski [Nonlinear Anal., to appear], to a class of optimal control problems in which constraint maps are also subject to variations. The main result of the paper is obtained as a realization of a variational principle extending the variational principle introduced in A. D. Ioffe and A. J. Zaslavski [SIAM J. Control Optim., 38 (2000), pp. 566--581].

Convergence of Krasnoselskii-Mann iterations of nonexpansive operators

In this paper, we show that a generic nonexpansive operator on a closed and convex, but not necessarily bounded, subset of a hyperbolic space has a unique fixed point which attracts the Krasnoselskii-Mann iterations of this operator.

STABLE CONVERGENCE THEOREMS FOR INFINITE PRODUCTS AND POWERS OF NONEXPANSIVE MAPPINGS

We show that several previously established convergence theorems for infinite products and powers of nonexpansive mappings continue to hold even when summable computational errors are present. Such results find application in methods for solving convex feasibility and optimization problems.