Minimax representation of nonexpansive functions and application to zero-sum recursive games

We show that a real-valued function on a topological vector space is positively homogeneous of degree one and nonexpansive with respect to a weak Minkowski norm if and only if it can be written as a minimax of linear forms that are nonexpansive with respect to the same norm. We derive a representation of monotone, additively and positively homogeneous functions on $L^\infty$ spaces and on $\mathbb{R}^n$, which extend results of Kolokoltsov, Rubinov, Singer, and others. We apply this representation to nonconvex risk measures and to zero-sum games. We derive in particular results of representation and polyhedral approximation for the class of Shapley operators arising from games without instantaneous payments (Everett's recursive games)

A B-convex production model for evaluating performance of firms

AbstractSome 30 years ago Charnes, Cooper and Rhodes [A. Charnes, W.W. Cooper, E. Rhodes, Measuring the efficiency of decision making units, European J. Oper. Res. 2 (6) (1978) 429–444] proposed DEA (Data Envelopement Analysis) as a mean of measuring and evaluating performance of firms. This paper proposes a model for production technologies which differs from the traditional DEA production model. The usual convex framework of the DEA model is replaced by an order theoretical condition: if two input vectors can produce a given output then the maximum coordinatewise of these two vectors can produce that same output. In this model, technologies are dually linked by a min–max cost function that is dual to the Shephard's distance function. Assuming free disposal of outputs these technologies can be completely described and the Shephard's distance function can be given in closed form

BEST APPROXIMATION IN QUASI TENSOR PRODUCT SPACE AND DIRECT SUM OF LATTICE NORMED SPACES

We study the theory of best approximation in tensor product and the direct sum of some lattice normed spacesX_{i}. We introduce quasi tensor product space anddiscuss about the relation between tensor product space and thisnew space which we denote it by X boxtimesY. We investigate best approximation in direct sum of lattice normed spaces by elements which are not necessarily downwardor upward and we call them I_{m}-quasi downward or I_{m}-quasi upward.We show that these sets can be interpreted as downward or upward sets. The relation of these sets withdownward and upward subsets of the direct sum of lattice normedspaces X_{i} is discussed. This will be done by homomorphismfunctions. Finally, we introduce the best approximation of thesesets

Best approximation by downward sets with applications

We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs (W,x), where x E X and W is a closed downward subset of X.C

Downward sets and their separation and approximation properties

We develop a theory of downward subsets of the space R-I, where I is a finite index set. Downward sets arise as the set of all solutions of a system of inequalities x is an element of R-I, f(t)(x) less than or equal to 0 (t is an element of T), where T is an arbitrary index set and each f(t) (t is an element of T) is an increasing function defined on R-I. These sets play an important role in some parts of mathematical economics and game theory. We examine some functions related to a downward set (the distance to this set and the plus-Minkowski gauge of this set, which we introduce here) and study lattices of closed downward sets and of corresponding distance functions. We discuss two kinds of duality for downward sets, based on multiplicative and additive min-type functions, respectively, and corresponding separation properties, and we give some characterizations of best approximations by downward sets. Some links between the multiplicative and additive cases are established