Dan Butnariu

Triangular Norm-Based Measures and Games with Fuzzy Coalitions

Introduction. I. Triangular Norm-Based Tribes. II. Triangular Norm-Based Measures. III. Too-Measures. IV. Games with Fuzzy Coalitions. V. Extensions of the Diagonal Value. VI. Related Topics and Applications. Bibliography. Index.

Totally convex functions for fixed points computation and infinite dimensional optimization

Introduction. 1. Totally Convex Functions. 2. Computation of Fixed Points. 3. Infinite Dimensional Optimization. Bibliography. Index.

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.

Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces

The aim of this paper is twofold. First, several basic mathematical concepts involved in the construction and study of Bregman type iterative algorithms are presented from a unified analytic perspective. Also, some gaps in the current knowledge about those concepts are filled in. Second, we employ existing results on total convexity, sequential consistency, uniform convexity and relative projections in order to define and study the convergence of a new Bregman type iterative method of solving operator equations.

Stability and Shapley value for an n-persons fuzzy game

Abstract The aim of the paper is to explain new concepts of solutions for n-persons fuzzy games. Precisely, it contains new definitions for ‘core’ and ‘Shapley value’ in the case of the n-persons fuzzy games. The basic mathematical results contained in the paper are these which assert the consistency of the ‘core’ and of the ‘Shapley value’. It is proved that the core (defined in the paper) is consistent for any n-persons fuzzy game and that the Shapley values exists and it is unique for any fuzzy game with proportional values.

Fixed points for fuzzy mappings

Abstract In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset Rχ of X. An element x of X is called fixed point of R iff its membership degree to Rχ is at least equal to the membership degree to Rχ of any y ϵ X, i.e. Rχ(χ)⩾ Rχ(y)(∀y ϵ X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of Rχ has a ‘complementary function’. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open.

Stable Convergence Behavior Under Summable Perturbations of a Class of Projection Methods for Convex Feasibility and Optimization Problems

We study the convergence behavior of a class of projection methods for solving convex feasibility and optimization problems. We prove that the algorithms in this class converge to solutions of the consistent convex feasibility problem, and that their convergence is stable under summable perturbations. Our class is a subset of the class of string-averaging projection methods, large enough to contain, among many other procedures, a version of the Cimmino algorithm, as well as the cyclic projection method. A variant of our approach is proposed to approximate the minimum of a convex functional subject to convex constraints. This variant is illustrated on a problem in image processing: namely, for optimization in tomography.

Iterative Averaging of Entropic Projections for Solving Stochastic Convex Feasibility Problems

The problem considered in this paper is that of finding a point which iscommon to almost all the members of a measurable family of closed convexsubsets of R++n, provided that such a point exists.The main results show that this problem can be solved by an iterative methodessentially based on averaging at each step the Bregman projections withrespect to f(x)=∑i=1nxi· ln xi ofthe current iterate onto the given sets.

Shapley mappings and the cumulative value for n-person games with fuzzy coalitions

Abstract In this paper we prove existence and uniqueness of the so-called Shapley mapping, which is a solution concept for a class of n -person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions. The Shapley mapping, when it exists, associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency, the symmetry, and the null-player conditions. It determines a “cumulative value” that is the “sum” of all coalitional allocations for whose computation we provide an explicit formula.

Additive fuzzy measures and integrals, III

Abstract A point of view concerning “fuzzy measures” is explained. To this end, a new concept of “disjointness” for fuzzy is introduced and studied. Also, a concept of an “additive class of fuzzy sets” is defined to be a class of fuzzy sets closed under some “additive operations.” The fuzzy measures are defined to be sum-preserving real functions over such additive classes. Some basic properties of the fuzzy measures are derived. In contrast with other homonymous concepts studied in literature, our fuzzy measures lead to an additive fuzzy integral (see the part II of the paper).