Projected Hessian Updating Algorithms for Nonlinearly Constrained Optimization

On the characterization of 𝑞-superlinear convergence of quasi-Newton methods for constrained optimization

“…In this note we have presented what we consider to be a short, direct and self-contained derivation of the Boggs-Tolle-Wang characterization of g-superlinear convergence for quasi-Newton methods for constrained optimization. While we have stated that the three previous derivations (Boggs, Tolle and Wang [1]; Fontecilla, Steihaug and Tapia [4] and Nocedal and Overton [7]) leave something to be desired, we quickly add that the present work was strongly influenced by these three papers. Indeed, the basic idea that led to the present derivation was to attempt to parallel the Nocedal-Overton derivation using a formulation of the quasi-Newton method which possessed the attribute that all necessary differentiations could be obtained in a straightforward manner.…”

Mise à jour de la métrique dans les méthodes de quasi-Newton réduites en optimisation avec contraintes d'égalité

“…Powell (1978), Gabay (1982b) et Nocedal, Overton (1985). Il s'obtient en négligeant dans le déplacement (1.12) de la PQS la partie du déplacement tangent qui fait intervenir l'approximation L k du hessien complet du lagrangien.…”

“…Un choix judicieux de ce critère permet d'obtenir la convergence superlinéaire de la suite (x k ). L'idée d'utiliser un critère de mise à jour a également été utilisée par Nocedal et Overton (1985) pour l'étude de l'algorithme (1.18).…”

Massively parallel aerodynamic shape optimization

“…On the other hand, the "long and thin" matrix Z T G^Y would increase storage requirements considerably over a path-following approach. Our approach is to ignore this term; Nocedal and Overton have shown that the resulting algorithm exhibits two-step Q-superlinear convergence [23], in the sense that…”