arXiv subjects

Walter Briec

Publications and source records attributed to Walter Briec.

9 recordsLinked to original sources

arXiv2025

On the Multiary Algebraic Formulation of an Idempotent Symmetric Limit Convex Structure

In [14], B-convexity was defined as an appropriate Painlev\'e-Kuratowski limit of linear convexities. More recently, an alternative algebraic formulation over the entire Euclidean vector space was proposed in [9] and [10]. The issue with the definition presented in [14] is that it was not developed from an algebraic perspective, but rather as the upper limit of a sequence of generalized convex polytopes whose form was not explicitly given. In this paper, we build on recent work and provide an algebraic formulation for these limiting polytopes. Consequently, we deduce a multiary algebraic form of B-convexity that involves an idempotent, non-associative algebraic structure, extending the formalism proposed in [9] to an arbitrary number of points. Among other things, we demonstrate that these limiting polytopes do not satisfy the idempotent symmetrical convex structure defined in [9]. In the context of this formalism, we derive a general separation result in Rn by approximating convex sets by polytopes. We conclude by clarifying some points regarding the external representation of polytopes proposed in [11] and analyze the structure of the polytopes that arise in this context.

math.OC↗

arXiv2024

Kolm-Pollack Form, Translation Homotheticity and Tropical Limit of Production Technologies

In this paper, we consider a new class of generalized Convex structure and we investigate their tropical limits. Some properties are pointing out such that translation homotheticity and others ones allowing to consider the case of discrete production sets that are related to some specific dual forms. Along this line a general class of mathematical programs are derived and it is shown that they can be computed using standard methods. The proposed approach allows to deal with efficiency measures (output oriented or input oriented) on continuous and discrete data.

math.OC↗

arXiv2024

Remarks on some Limit Geometric Properties related to an Idempotent and Non-Associative Algebraic Structure

This article analyzes the geometric properties of an idempotent, non-associative algebraic structure that extends the Max-Times semiring. This algebraic structure is useful for studying systems of Max-Times and Max-Plus equations, employing an appropriate notion of a non-associative determinant. We consider a connected ultrametric distance and demonstrate that it implies, among other properties, an analogue of the Pythagorean relation. To this end, we introduce a suitable notion of a right angle between two vectors and investigate a trigonometric concept associated with the Chebyshev unit ball. Following this approach, we explore the potential implications of these properties in the complex plane. We provide an algebraic definition of a line passing through two points, which corresponds to the Painlev\'e-Peano-Kuratowski limit of a sequence of generalized lines. We establish that this definition leads to distinctive geometric properties; in particular, two distinct parallel lines may share an infinite number of points.

math.RA↗

arXiv2022

$\Lambda$-Returns to Scale and Individual Minimum Extrapolation Principle

This paper proposes to estimate the returns-to-scale of production sets by considering the individual return of each observed firm through the notion of $\Lambda$-returns to scale assumption. Along this line, the global technology is then constructed as the intersection of all the individual technologies. Hence, an axiomatic foundation is proposed to present the notion of $\Lambda$-returns to scale. This new characterization of the returns-to-scale encompasses the definition of $\alpha$-returns to scale, as a special case as well as the standard non-increasing and non-decreasing returns-to-scale models. A non-parametric procedure based upon the goodness of fit approach is proposed to assess these individual returns-to-scale. To illustrate this notion of $\Lambda$-returns to scale assumption, an empirical illustration is provided based upon a dataset involving 63 industries constituting the whole American economy over the period 1987-2018.

econ.GN↗

arXiv2021

Distance Functions and Generalized Means: Duality and Taxonomy

This article demonstrates how a large number of efficiency measures known in the literature in production economics can be interpreted through the notion of utility function, based on the concept of Stone-Geary utility. Several relationships between these utility functions and distance functions, a commonly used tool in production theory, are established. To achieve these objectives, a generalized mean distance function is introduced, inspired by the Atkinson inequality index, itself derived from the notion of the Aczel mean. It measures the maximum sum of netput expansions required to reach an efficient point. Several duality theorems are established, linking the new distance functions to the profit function. For all feasible production vectors, the results include as special cases most of the dual correspondences previously established in the literature. Finally, a large class of measures is identified for which these duality results can be obtained without requiring convexity. A numerical example is provided.

econ.TH↗

arXiv2020

Determinants and Limit Systems in some Idempotent and Non-Associative Algebraic Structure

This paper considers an idempotent and symmetrical algebraic structure as well as some closely related concept. A special notion of determinant is introduced and a Cramer formula is derived for a class of limit systems derived from the Hadamard matrix product and we give the algebraic form of a sequence of hyperplanes passing through a finite number of points. Thereby, some standard results arising for Max-Times systems with nonnegative entries appear as a special case. The case of two sided systems is also analyzed. In addition, a notion of eigenvalue in limit is considered. It is shown that one can construct a special semi-continuous regularized polynomial to find the eigenvalues of a matrix with nonnegative entries.

math.CO↗

arXiv2013

On Some Idempotent and Non-Associative Convex Structure

$\mathbb B$-convexity was defined in [7] as a suitable Kuratowski-Painlevé upper limit of linear convexities over a finite dimensional Euclidean vector space. Excepted in the special case where convex sets are subsets of $\mathbb R^n_ +$, $\mathbb B$-convexity was not defined with respect to a given explicit algebraic structure. This is done in that paper, which proposes an extension of $\mathbb B$-convexity to the whole Euclidean vector space. An unital idempotent and non-associative magma is defined over the real set and an extended $n$-ary operation is introduced. Along this line, the existence of the Kuratowski-Painlevé limit of the convex hull of two points over $\mathbb R^n$ is shown and an explicit extension of $\mathbb B$-convexity is proposed.

math.OC↗

arXiv2011

Quasi-Leontief utility functions on partially ordered sets I: efficient points

A function $u: X\to\mathbb{R}$ defined on a partially ordered set is quasi-Leontief if, if for all $x\in X$, the upper level set $\{x^\prime\in X: u(x^\prime)\geqslant u(x)\} $ has a smallest element. A function $u: \prod_{j=1}^nX_j\to\mathbb{R}$ whose partial functions obtained by freezing $n-1$ of the variables are all quasi-Leontief is an individually quasi-Leontief function; a point $x$ of the product space is an efficient point for $u$ if it is a minimal element of $\{x^\prime\in X: u(x^\prime)\geqslant u(x)\} $. Part I deals with the maximisation of quasi-Leontief functions and the existence of efficient maximizers. Part II is concerned with the existence of efficient Nash equilibria for abstract games whose payoff functions are individually quasi-Leontief. Order theoretical and algebraic arguments are dominant in the first part while, in the second part, topology is heavily involved. In the framework and the language of tropical algebras, our quasi-Leontief functions are the additive functions defined on a semimodule with values in the semiring of scalars.

math.OC↗

arXiv2011

Quasi-Leontief utility functions on partially ordered sets II: Nash equilibria

We prove that, under appropriate conditions, an abstract game with quasi-Leontief payoff functions $u_i : \prod_{j=1}^nX_j\to\mathbb{R}$ has a Nash equilibria. When all the payoff functions are globally quasi-Leontief, the existence and the characterization of efficient Nash equilibria mainly follows from the analysis carried out in part I. When the payoff functions are individually quasi-Leontief functions the matter is somewhat more complicated. We assume that all the strategy spaces are compact topological semilattices, and under appropriate continuity conditions on the payoff functions, we show that there exists an efficient Nash equilibria using the Eilenberg-Montgomery Fixed Point Theorem for acyclic valued upper semicontinuous maps defined on an absolute retract and some non trivial properties of topological semilattices. The map in question is defined on the set of Nash equilibria and its fixed points are exactly the efficient Nash equilibria.

math.OC↗