Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Majorant ou minorant
Résumé
In mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K that is greater than or equal to every element of S.
Dually, a lower bound or minorant of S is defined to be an element of K that is less than or equal to every element of S.
A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound.
The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.
For example, 5 is a lower bound for the set S = (as a subset of the integers or of the real numbers, etc.), and so is 4. On the other hand, 6 is not a lower bound for S since it is not smaller than every element in S.
The set S = has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S.
Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above. An infinite subset of the integers may be bounded from below or bounded from above, but not both. An infinite subset of the rational numbers may or may not be bounded from below, and may or may not be bounded from above.
Every finite subset of a non-empty totally ordered set has both upper and lower bounds.
The definitions can be generalized to functions and even to sets of functions.
Given a function with domain D and a preordered set (K, ≤) as codomain, an element y of K is an upper bound of if y ≥ (x) for each x in D. The upper bound is called sharp if equality holds for at least one value of x. It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality.
Similarly, a function g defined on domain D and having the same codomain (K, ≤) is an upper bound of , if g(x) ≥ (x) for each x in D.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Proximité ontologique
Cours associés (14)
CS-439: Optimization for machine learning
This course teaches an overview of modern optimization methods, for applications in machine learning and data science. In particular, scalability of algorithms to large datasets will be discussed in t
MATH-101(en): Analysis I (English)
We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable.