A Newton–Secant method for differentiable set-valued maps in Nachi–Penot sense

Gaydu, Michaël ; Yacinthe, Olguine ; Nuiro, Silvere Paul ; Pietrus, Alain

Serdica Mathematical Journal, Tome 48 (2022) no. 1–2, pp. 41-68.

Résumé

This paper is concerned with the solving of variational inclusions of the form \(0\in f(x) + g(x) + F(x) – K\), where \(g\) is a function which is differentiable at a solution \(x^{*}\) of the inclusion but may be not differentiable in a neighborhood of \(x^{*}\). The function \(f\) and the set-valued mapping \(F\) are differentiable in the sense of Nachi–Penot [14] and \(K\) is a nonempty closed convex cone.We introduce a Newton-Secant method to solve our problem and the sequence associated is semilocally convergent to \(x^{*}\) with an order equal to \(\frac{1 +\sqrt{5}}{2}\). Finally, some numerical results are also given to illustrate the convergence of the proposed method.

Détail

Keywords: variational inclusions, set-valued maps, divided differences, generalized differentiation of set-valued maps, normed convex processes, majorizing sequences, 49J53, 49J40, 90C48, 65K10