Newton's method for solving generalized equations: Kantorovich's and Smale's approaches

Samir Adly (1) , Huynh van Ngai (2) , van Vu Nguyen (1)

1 XLIM-MATHIS - Mathématiques & Sécurité de l'information
2 Department of Mathematics, Pedagogical University of Quynhon

In this paper, we study Newton-type methods for solving generalized equations involving set-valued maps in Banach spaces. Kantorovich-type theorems (both local and global versions) are proved as well as the quadratic convergence of the Newton sequence. We also extend Smale's classical (α,γ) -theory to generalized equations. These results are new and can be considered as an extension of many known ones in the literature for classical nonlinear equations. Our approach is based on tools from variational analysis. The metric regularity concept plays an important role in our analysis.

Mots clés: Generalized equation; Newton's method; Metric regularity; Kantorovich's theorem; Smale's ($\alpha-\gamma$)-theory; Variational analysis

Domaines: Optimisation et contrôle [math.OC]

Samir Adly, Huynh van Ngai, van Vu Nguyen. Newton's method for solving generalized equations: Kantorovich's and Smale's approaches. Journal of Mathematical Analysis and Applications, 2016, 439 (1), pp.396-418. ⟨10.1016/j.jmaa.2016.02.047⟩. ⟨hal-01313209⟩