Metric Regularity of Newton's Iteration

Artacho, Francisco J. Aragón; Dontchev, Asen L.; Gaydu, Michaël; Geoffroy, Michel; Veliov, Vladimir M.

doi:10.1137/100792585

Cited by 48 publications

(31 citation statements)

References 12 publications

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“…Under assumptions (α), (β) and (γ ) Azé and Chou generate a Newton sequence x n satisfying the above assertions (1)- (3) and strongly converging to a solution to the inclusion (36).…”

Section: Discussionmentioning

confidence: 99%

“…Additional results in connection with this Newton iteration for generalized equations can be found in [3,4]. It was also in the mid-nineteen nineties that Azé and Chou [5] presented a Newton method for solving the inclusion…”

Section: Introductionmentioning

confidence: 88%

“…Dontchev showed the local quadratic convergence of a Newton-type iteration, based on a partial linearization of the mapping f + F , for solving (3).…”

Section: Introductionmentioning

confidence: 99%

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A Newton iteration for differentiable set-valued maps

Gaydu

Geoffroy

2013

Journal of Mathematical Analysis and Applications

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a b s t r a c tWe employ recent developments of generalized differentiation concepts for set-valued mappings and present a Newton-like iteration for solving generalized equations of the formboth of them being smooth mappings acting between two general Banach spaces X and Y . The Newton iteration we propose is constructed on the basis of a linearization of both f and F ; we prove that, under suitable assumptions on the ''derivatives'' of f and F , it converges Q-linearly to a solution to the generalized equation in question. When we strengthen our assumptions, we obtain the Q-quadratic convergence of the method.

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“…Under assumptions (α), (β) and (γ ) Azé and Chou generate a Newton sequence x n satisfying the above assertions (1)- (3) and strongly converging to a solution to the inclusion (36).…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 88%

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A Newton iteration for differentiable set-valued maps

Gaydu

Geoffroy

2013

Journal of Mathematical Analysis and Applications

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“…This method may be viewed as a Newton-type method based on a partial linearization, which has been studied in several papers including [1,2,8,12]; see also [10,Section 6C]. When F ≡ 0, the iteration (2) becomes the standard Newton's method for solving the nonlinear equation f (x) = 0. If X = R n , Y = R m and F = R s − × {0} m−s , then (2) is a Newton's method for solving a system of equalities and inequalities; see [6].…”

Section: Introductionmentioning

confidence: 99%

Kantorovich's Theorem on Newton's Method for Solving Strongly Regular Generalized Equation

Ferreira¹,

Silva²

2017

SIAM J. Optim.

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In this paper we consider the Newton's method for solving the generalized equation of the form f (x) + F (x) ∋ 0, where f : Ω → Y is a continuously differentiable mapping, X and Y are Banach spaces, Ω ⊆ X an open set and F : X ⇒ Y be a set-valued mapping with nonempty closed graph. We show that, under strong regularity of the generalized equation, concept introduced by S. M. Robinson in [27], and starting point satisfying the Kantorovich's assumptions, the Newton's method is quadratically convergent to a solution, which is unique in a suitable neighborhood of the starting point. The analysis presented based on Banach Perturbation Lemma for generalized equation and the majorant technique, allow to unify some results pertaining the Newton's method theory.1 known version of the Newton's method for solving variational inequality; see [7,18]. In particular, if (1) represents the Karush-Kuhn-Tucker optimality conditions for a mathematical programming problem, then the procedure (2) describes the well-known sequential quadratic programming method; see for example [10, pag. 334].L. V. Kantorovich in [19], see also [20,23], was the first to prove a convergence result for Newton's method for solving the equation f (x) = 0, where f : Ω → Y is a continuously differentiable mapping, X and Y are Banach spaces and Ω ⊆ X is an open set. Using conditions on x 0 the starting point, namely, under the condition that f ′ (x 0 ) −1 exists and f ′ (x 0 ) −1 f (x 0 ) is bounded, L.V. Kantorovich obtained well definition of the method, quadratic convergence and uniqueness of solution. The idea employed in the proof of convergence was the technique of majorization, which consists in bound the Newton's sequence by a scalars sequence. This technique has been used and extended for various researchers, including [5,13,15,16,17,24,30,32]. S. M. Robinson in [25], using the idea of convex process introduced by Rockafellar [29], see also [26,28], established a generalization of the Kantorovich's theorem for solving the inclusion f (x) ∈ C, where f : Ω → Y is a continuously differentiable mapping, X and Y are Banach spaces, Ω ⊆ X is an open set and C ⊆ Y is a nonempty closed and convex cone. The paper [25] has been extended for various authors, see for instance [5,13,15,21]. In his Ph.D. thesis, N. H. Josephy in [18] studied Newton's method for solving the variational inequality f (x) + N C ∋ 0, where f : Ω → R m is a continuously differentiable mapping, Ω ⊆ R n is an open set and N C is the normal cone mapping of a convex set C ⊂ R m . For guarantee the well definition of the method, strong regularity property on f (x) + N C , concept introduced in the theory of generalized equations by S.M. Robinson in [27], was used. If X = Y and N C = {0}, then strong regularity is equivalent to f ′ (x) −1 be a continuous linear operator. If X = R n , Y = R m and F = R s − × {0} m−s , then strong regularity is equivalent to Mangasarian-Fromovitz constraint qualification; see [10, Example 4D.3]. An important case is when (1) represents the Karush-Kuhn-Tucker's ...

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“…We will follow the same idea from [4,9], where the authors extend the paradigm of the Lyusternik-Graves theorem (see, e.g., [8]) to the framework of a mapping acting from the pair initial point-parameter to the set of convergent Newton sequences associated with them. Under some surjectivity assumption, known as ample parameterization, the (strong) metric regularity of the generalized equation is proved to be equivalent to the (strong) metric regularity of the inverse mapping associated with convergent Newton sequences.…”

Section: Introductionmentioning

confidence: 99%

A Lyusternik–Graves theorem for the proximal point method

Artacho

Gaydu

2011

Comput Optim Appl

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We consider a generalized version of the proximal point algorithm for solving the perturbed inclusion y ∈ T (x), where y is a perturbation element near 0 and T is a set-valued mapping acting from a Banach space X to a Banach space Y which is metrically regular around some point point (x, 0) in its graph. We study the behavior of the convergent iterates generated by the algorithm and we prove that they inherit the regularity properties of T , and vice versa. We analyze the cases when the mapping T is metrically regular and strongly regular.

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Metric Regularity of Newton's Iteration

Cited by 48 publications

References 12 publications

A Newton iteration for differentiable set-valued maps

A Newton iteration for differentiable set-valued maps

Kantorovich's Theorem on Newton's Method for Solving Strongly Regular Generalized Equation

A Lyusternik–Graves theorem for the proximal point method

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