Direct Search Based on Probabilistic Descent

Abstract : Direct-search methods are a class of popular derivative-free algorithms characterized by evaluating the objective function using a step size and a number of (polling) directions. When applied to the minimization of smooth functions, the polling directions are typically taken from positive spanning sets, which in turn must have at least n+1 vectors in an $n$-dimensional variable space. In addition, to ensure the global convergence of these algorithms, the positive spanning sets used throughout the iterations are required to be uniformly nondegenerate in the sense of having a positive (cosine) measure bounded away from zero. However, recent numerical results indicated that randomly generating the polling directions without imposing the positive spanning property can improve the performance of these methods, especially when the number of directions is chosen as considerably less than n+1. In this paper, we analyze direct-search algorithms when the polling directions are probabilistic descent, meaning that with a certain probability at least one of them is of descent type. Such a framework enjoys almost-sure global convergence. More interestingly, we will show a global decaying rate of $1/\sqrt{k}$ for the gradient size, with overwhelmingly high probability, matching the corresponding rate for the deterministic versions of the gradient method or of direct search. Our analysis helps us understand numerical behavior and the choice of the number of polling directions.
Type de document :
Article dans une revue
SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2015, vol. 25 (n° 3), pp. 1515-1541. <10.1137/140961602>
Liste complète des métadonnées


https://hal.archives-ouvertes.fr/hal-01523690
Contributeur : Open Archive Toulouse Archive Ouverte (oatao) <>
Soumis le : mardi 16 mai 2017 - 16:57:39
Dernière modification le : vendredi 19 mai 2017 - 01:06:02

Fichier

gratton_16987.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

Collections

Citation

Serge Gratton, Clément Royer, Luis Vicente, Zaikun Zhang. Direct Search Based on Probabilistic Descent. SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2015, vol. 25 (n° 3), pp. 1515-1541. <10.1137/140961602>. <hal-01523690>

Partager

Métriques

Consultations de
la notice

14

Téléchargements du document

13