Theories of Fixed Point Index and Applications

This thesis is devoted to the study of theories of fixed point index for generalized and weakly inward maps of condensing type and weakly inward A-proper maps. In Chapter 1 we recall some basic concepts such as cones, wedges, measures of noncompactness and theories of fixed point index for compact and gamma-condensing self-maps. We also give some new results and provide new proofs for some known results. In Chapter 2 we study approximatively compact sets giving examples and proving new results. The concept of an approximatively compact set is of importance in defining our index for a generalized inward map since there exists upper semicontinuous multivalued metric projections onto the approximatively compact convex set. We also introduce the concept of an M1-set which will play an important role in defining our fixed point index for generalized inward maps of condensing type since there exists continuous single-valued metric projections onto an Ml-closed convex set. Many examples of M1-closed convex sets are given. Weakly inward sets and weakly inward maps are studied in detail. New properties and examples on such sets and maps are given. We also introduce the new concept of generalized inward sets and generalized inward maps. The class of generalized inward maps strictly contain the class of weakly inward maps. Several necessary and sufficient conditions for a map to be generalized inward and examples of generalized inward maps are given. In Chapter 3 we define a fixed point index for a generalized inward compact map defined on an approximatively compact convex set and obtain many new fixed point theorems and nonzero fixed point theorems. In particular, norm-type expansion and compression theorems for weakly inward continuous maps in finite dimensional Banach spaces are obtained, which have not been considered previously. In Chapter 4 we define a fixed point index for a generalized inward maps of condensing type defined on an M1-closed convex set and obtain many new fixed point theorems and nonzero fixed point theorems. We also apply the abstract theory to some perturbed Volterra equations. In Chapter 5 we define a fixed point index for weakly inward A-proper maps. We obtain new fixed point theorems, nonzero fixed point theorem and results on existence of eigenvalues. We also give an application of the abstract theory to the existence of nonzero positive solutions of boundary value problems for second order differential equations

Projections Onto Convex Sets (POCS) Based Optimization by Lifting

Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex and some non-convex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function is a convex function in R^N the corresponding set is a convex set in R^(N+1). The iterative optimization approach starts with an arbitrary initial estimate in R^(N+1) and an orthogonal projection is performed onto one of the sets in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in total-variation, filtered variation, l1, and entropic cost functions. It is also experimentally observed that cost functions based on lp, p<1 can be handled by using the supporting hyperplane concept

Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods

The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projection method is perturbation resilient. By this we mean that under appropriate conditions the sequence generated by the method converges weakly, and sometimes also strongly, to a point in the intersection of the given subsets of the feasibility problem, despite certain perturbations which are allowed in each iterative step. Unlike previous works on solving the convex feasibility problem, the involved functions, which induce the feasibility problem's subsets, need not be convex. Instead, we allow them to belong to a wider and richer class of functions satisfying a weaker condition, that we call "zero-convexity". This class, which is introduced and discussed here, holds a promise to solve optimization problems in various areas, especially in non-smooth and non-convex optimization. The relevance of this study to approximate minimization and to the recent superiorization methodology for constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio

Optimal Convergence Rates for Generalized Alternating Projections

Generalized alternating projections is an algorithm that alternates relaxed projections onto a finite number of sets to find a point in their intersection. We consider the special case of two linear subspaces, for which the algorithm reduces to a matrix teration. For convergent matrix iterations, the asymptotic rate is linear and decided by the magnitude of the subdominant eigenvalue. In this paper, we show how to select the three algorithm parameters to optimize this magnitude, and hence the asymptotic convergence rate. The obtained rate depends on the Friedrichs angle between the subspaces and is considerably better than known rates for other methods such as alternating projections and Douglas-Rachford splitting. We also present an adaptive scheme that, online, estimates the Friedrichs angle and updates the algorithm parameters based on this estimate. A numerical example is provided that supports our theoretical claims and shows very good performance for the adaptive method.Comment: 20 pages, extended version of article submitted to CD

Successive Radii and Ball Operators in Generalized Minkowski Spaces

We investigate elementary properties of successive radii in generalized Minkowski spaces (that is, with respect to gauges), i.e., we measure the "size" of a given convex set in a finite-dimensional real vector space with respect to another convex set. This is done via formulating some kind of minimal containment problems, where intersections or Minkowski sums of the latter set and affine flats of a certain dimension are incorporated. Since this is strongly related to minimax location problems and to the notions of diametrical completeness and constant width, we also have a look at ball intersections and ball hulls.Comment: submitted to "Advances of Geometry

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques