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
Pseudoconvex function
Résumé
In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has a positive directional derivative. The property must hold in all of the function domain, and not only for nearby points.
Consider a differentiable function , defined on a (nonempty) convex open set of the finite-dimensional Euclidean space . This function is said to be pseudoconvex if the following property holds:
Equivalently:
Here is the gradient of , defined by:
Note that the definition may also be stated in terms of the directional derivative of , in the direction given by the vector . This is because, as is differentiable, this directional derivative is given by:
Every convex function is pseudoconvex, but the converse is not true. For example, the function is pseudoconvex but not convex. Similarly, any pseudoconvex function is quasiconvex; but the converse is not true, since the function is quasiconvex but not pseudoconvex. This can be summarized schematically as:
To see that is not pseudoconvex, consider its derivative at : . Then, if was pseudoconvex, we should have:
In particular it should be true for . But it is not, as: .
For any differentiable function, we have the Fermat's theorem necessary condition of optimality, which states that: if has a local minimum at in an open domain, then must be a stationary point of (that is: ).
Pseudoconvexity is of great interest in the area of optimization, because the converse is also true for any pseudoconvex function. That is: if is a stationary point of a pseudoconvex function , then has a global minimum at . Note also that the result guarantees a global minimum (not only local).
This last result is also true for a convex function, but it is not true for a quasiconvex function. Consider for example the quasiconvex function:
This function is not pseudoconvex, but it is quasiconvex.
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.