0000000000147681
AUTHOR
Roland Durier
Sets of Efficiency in a Normed Space and Inner Product
In a normed space X the distances to the points of a given set A being considered as the objective functions of a multicriteria optimization problem, we define four sets of efficiency (efficient, strictly efficient, weakly efficient and properly efficient points). Instead of studying properties of the sets of efficiency according to properties of the norm, we investigate an inverse problem: deduce properties of the norm of X from properties of the sets of efficiency, valid for every finite subset A of X.
Optimal Locations and Inner Products
Abstract In a normed space X , we consider objective functions which depend on the distances between a variable point and the points of certain finite sets A . A point where such a function attains its minimum on X is generically called an optimal location. In this paper we obtain characterizations of inner product spaces with properties connecting optimal locations and the convex hull of A or barycenters of points of A with well chosen weights. We thus generalize several classical results about characterization of inner product spaces.
Constrained and unconstrained problems in location theory and inner products
In a real normed space X the optimization problem associated to a finite subset and to a family of positive weights with the objective function [UM0001] has some well known properties when X is an ...
A General Framework for the One Center Location Problem
This paper deals with an optimization problem where the objective function F is defined on a real vector space X by F(x) = γ(w 1║x - a 1║1, ⋯, w n ║x - a n║ n ), a formula in which a 1, ⋯, a n are n given points in X, ║∙║1, ⋯, ║∙║ n n norms on X, w 1, ⋯, w n positive numbers and γ a monotone norm on ℝ n . A geometric description of the set of optimal solutions to the problem min F(x) is given, illustrated by some examples. When all norms ║∙║i are equal, and γ being successively the l 1 , l ∞ and l 2-norm, a particular study is made, which shows the peculiar role played by the l 1-norm.