Extended Semismooth Newton Method for Functions with Values in a Cone
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Extended Semismooth Newton Method for Functions with Values in a Cone Séverine Bernard1 · Catherine Cabuzel1 · Silvère Paul Nuiro1 · Alain Pietrus1
Received: 13 May 2016 / Accepted: 26 November 2017 © Springer Science+Business Media B.V., part of Springer Nature 2017
Abstract This paper deals with variational inclusions of the form 0 ∈ K − f (x) where f : Rn → Rm is a semismooth function and K is a nonempty closed convex cone in Rm . We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of f . The results obtained in this paper extend those obtained by Robinson in the famous paper (Robinson in Numer. Math. 19:341–347, 1972). We provide a semilocal method with a superlinear convergence that is new in the context of semismooth functions. Finally, numerical results are also given to illustrate the convergence. Keywords Variational inclusion · Semismooth function · Closed convex cone · Majorizing sequence · Normed convex process Mathematics Subject Classification 49J53 · 47H04 · 65K10 · 14P15
1 Introduction The variational inclusions were introduced by Robinson [33, 34] as an abstract model for various problems encountered in different fields such as engineering, economy, transport theory, etc. Several studies dealing with the variational inclusions of the form: 0 ∈ f (x) + F (x).
B A. Pietrus
[email protected] S. Bernard [email protected] C. Cabuzel [email protected] S.P. Nuiro [email protected]
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Laboratoire LAMIA, EA 4540, Département de Mathématiques et Informatique, Université des Antilles, Campus de Fouillole, 97159 Pointe-à-Pitre, France
(1)
S. Bernard et al.
have been carried out during the last decades where f : X → Y is a function; F : X ⇒ Y is a set-valued map and X, Y are Banach spaces. In the smooth case (f is smooth), Dontchev [12, 13] gave an interesting contribution to approximate a solution x ∗ for (1). For this, he introduced a sequence obtained from a partial linearization of the single-valued part. More precisely, he associated the Newtontype sequence to (1) x0 is a given starting point (2) 0 ∈ f (xk ) + f (xk )(xk+1 − xk ) + F (xk+1 ). and established the quadratic convergence when f (the Fréchet derivative of f ) is locally Lipschitz around the solution x ∗ and under a pseudo-Lipschitz property for the set-valued map (f + F )−1 . For more details on the Lipschitz property, also called Aubin property or Lipschitz like property, the reader could refer to [2, 3, 14, 15, 24, 25, 36, 37]. Following Dontchev’s works, we can find various papers in the literature about the resolution of this type of variational inclusions in which the authors use metric regularity. It seems natural to study the nonsmooth case. In this case, Cabuzel and Piétrus [6] introduced a similar method for subanalytic functions replacing in (2), f (xk ) by f (xk ) taken in the Clarke generalized Jacobian of f at xk . The concerned sequence is thus x0 is a given starting point (3) 0 ∈ f (xk ) + f (xk )(xk+1 − xk ) + F (xk+1 ). In their paper,
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