Ph.D Student Gabour Manal Some Methods for Solving Convex Feasibility and Optimization Problems Department of Applied Mathematics Professor Emeritus Simeon Reich

Abstract

The Convex Feasibility Problem is the problem of finding a point in the intersection of a family of closed convex sets in a Banach space. This problem is of broad interdisciplinary interest in many areas of applied mathematics and engineering. In mathematical programming it is usually stated as the problem of finding a solution to a system of nonlinear inequalities. It also arises in set theoretic design, estimation problems, transportation, antenna array design, communications, topography, digital filter design and image recovery problems. The methods we use for dealing with the convex feasibility problem involve the iterations of nonexpansive mappings, namely the Parallel Iteration Method, The Random Product Method and the Product Space Method. It turns out that in many of the applied problems mentioned above, inconsistencies may arise as a result of noisy measurements or inaccurate constraints. The problem of finding common fixed points of nonexpansive retractions defined on a Banach space onto closed convex subsets of it is considered, and in the case where all the given sets are bounded, we obtain a result on the existence of fixed points. We  also have a weak convergence result, to fixed points, of the parallel algorithm iterations. In the case where only one of the sets is bounded, we show that the iterations of the parallel algorithm are bounded. By imposing some conditions on the space, we obtain an existence result for fixed points. These results are also applicable to the inconsistent case mentioned above.

A generalized stochastic convex feasibility problem is also considered. This is the problem of finding almost common fixed points of measurable families of nonexpansive mappings . A generic approach is used to study the convergence of iterates and of infinite products. Instead of considering a certain convergence property for powers of a single operator or for a single sequence of operators, an investigation is made regarding a space of all such operators or sequences equipped with some natural complete metric and it is shown that this property holds for most of these sequences. More precisely, it is shown that in appropriate spaces of operators or sequences of operators there exists a subset which is a countable intersection of open everywhere dense sets such that for each operator or sequence belonging to this subset, the corresponding sequence of powers or infinite product converges. Generic strong convergence results are established. One important point in these strong convergence results is that they are also applicable to the inconsistent case, where there are no common fixed points.

The generic approach is also applied to a different problem. The concept of normality is introduced for (sequences of) self-mappings of a nonempty, bounded, closed and convex subset of a Banach space. These self-mappings share a common convex uniformly continuous Lyapunov function. A complete metric space of sequences of such self-mappings is defined and it is shown that a generic element taken from this space is normal. This result is important because for such an element, the sequence of values of the Lyapunov function along any trajectory of it, tends to the infimum of the function. From the point of view of the theory of dynamical systems, each element of our metric space describes a nonstationary dynamical system with a Lyapunov function. Also, some optimization procedures in Banach spaces can be represented by elements of that metric space.