Application des méthodes d'optimisation pour la résolution du problème d'inégalités variationnelles
Application of optimization methods for solving the problem of variational inequalities
DOI:
https://doi.org/10.33993/jnaat361-859Keywords:
problème d'inégalités variationnelles, fonctions de mérite, optimisation sans contraintes, méthode de premier ordre, méthode de second ordreAbstract
In French.
Le problème d'inégalités variationnelles, lancé au milieu des années soixantes par Hartman et Stampacchia dans le cadre du calcul des variations, et celui des problèmes aux limites des équations aux dérivées partielles, prend depuis quelques années une importance grandissante dans l'étude théorique et le traitement numérique de plusieurs types de problèmes pratiques et scientifiques d'intérêt capital.
Les techniques d'optimisation constituent un bon stimulant conduisant à une méthodologie riche et pleine d'expériences. A ce propos, ce problème est converti en un problème d'optimisation équivalent, avec ou sans contraintes, ayant les propriétés requises pour un traitement convenable.
Notre travail se rattache à des méthodes d'optimisation sans contraintes. Nous avons pu mettre en oeuvre plusieurs versions de ces algorithmes présentées dans un cadre comparatif signifiant, à travers des problèmes mathématiques importants.
In English.
The problem of variational inequalities, launched in the mid-sixties by Hartman and Stampacchia in the context of the calculus of variations, and that of problems at the limits of partial differential equations, has become increasingly important in recent years in the theoretical study and numerical treatment of several types of practical and scientific problems of capital interest.
Optimization techniques are a good stimulus leading to a rich and experienced methodology. In this regard, this problem is converted into an equivalent optimization problem, with or without constraints, having the properties required for a suitable treatment.
Our work is related to optimization methods without constraints. We were able to implement several versions of these algorithms presented in a meaningful comparative framework, through important mathematical problems.
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