Projection Algorithms for Solving the Split Feasibility Problem with Multiple Output Sets

Journal of Optimization Theory and Applications - We study the split common fixed point problem for Bregman relatively nonexpansive operators and the split feasibility problem with multiple output...     

Re-examination of Bregman functions and new properties of their divergences

ABSTRACT

The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the ‘Bregman function’). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, operations research, information theory, nonlinear analysis, machine learning and more. This paper re-examines various aspects related to the theory of Bregman functions and divergences. In particular, it presents many sufficient conditions which allow the construction of Bregman functions in a general setting and introduces new Bregman functions (such as a negative iterated log entropy). Moreover, it sheds new light on several known Bregman functions such as quadratic entropies, the negative Havrda-Charvát-Tsallis entropy, and the negative Boltzmann-Gibbs-Shannon entropy, and it shows that the negative Burg entropy, which is not a Bregman function according to the classical theory but nevertheless is known to have ‘Bregmanian properties’, can, by our re-examination of the theory, be considered as a Bregman function. Our analysis yields several by-products of independent interest such as the introduction of the concept of relative uniform convexity (a certain generalization of uniform convexity), new properties of uniformly and strongly convex functions, and results in Banach space theory.     

Linear fractional mappings: invariant sets, semigroups and commutativity

We study commutativity and embeddability (into continuous semi-groups) properties of linear fractional self-mappings of the open unit disk in the complex plane. The common thread in our approach is the classical notion of the Kœnigs function which we use in each of the three possible cases (dilation, hyperbolic and parabolic). Since we are interested in a classical subject, the paper is written in the style of a survey, in order to make it accessible to a wider audience. Therefore it contains, in addition to our new results, an exposition of most relevant facts. Dedicated to Professor Felix E. Browder with admiration and respect     

Algorithms for the Split Variational Inequality Problem

We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails finding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.     

The equivalence of linear codes implies semi-linear equivalence

Designs, Codes and Cryptography - We prove that if two linear codes are equivalent then they are semi-linearly equivalent. We also prove that if two additive MDS codes over a field are equivalent...     

Contributions by Aart Blokhuis to finite geometry,discrete mathematics,and combinatorics

Designs, Codes and Cryptography -     

Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem

We present a new proof of the classical Kirszbraun-Valentine extension theorem. Our proof is based on the Fenchel duality theorem from convex analysis and an analog for nonexpansive mappings of the Fitzpatrick function from monotone operator theory.