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G N Silva - One of the best experts on this subject based on the ideXlab platform.
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kantorovich s theorem on newton s method for solving strongly regular Generalized Equation
Siam Journal on Optimization, 2017Co-Authors: O P Ferreira, G N SilvaAbstract:In this paper, we consider Newton's method for solving a Generalized Equation. We show that under strong regularity of the Equation and on the condition that the starting point satisfies Kantorovich's assumptions, the method is quadratically convergent to a solution, which is unique within a suitable neighborhood of the starting point. Our analysis, which is based on Kantorovich's majorant technique, enables us to obtain a convergence result under the Lipschitz, Smale, and Nesterov--Nemirovskii self-concordant conditions.
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Local convergence of Newton’s method for solving Generalized Equations with monotone operator
Applicable Analysis, 2017Co-Authors: G N SilvaAbstract:In this paper, we study Newton’s method for solving the Generalized Equation in Hilbert spaces, where F is a Frechet differentiable function and T is set-valued and maximal monotone. We show that this method is locally quadratically convergent to a solution. Using the idea of a majorant condition on the nonlinear function, which is associated with the Generalized Equation, the convergence of the method, the optimal convergence radius, and results of the convergence rate are established. The advantage of working with a majorant condition rests in the fact that it allows unifying of several convergence results pertaining to Newton’s method.
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kantorovich s theorem on newton s method for solving strongly regular Generalized Equation
arXiv: Numerical Analysis, 2016Co-Authors: O P Ferreira, G N SilvaAbstract:In this paper we consider the Newton's method for solving the Generalized Equation of the form $ f(x) +F(x) \ni 0, $ where $f:{\Omega}\to Y$ is a continuously differentiable mapping, $X$ and $Y$ are Banach spaces, $\Omega\subseteq X$ an open set and $F:X \rightrightarrows Y$ 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.
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Local convergence analysis of Newton's method for solving strongly regular Generalized Equations
arXiv: Numerical Analysis, 2016Co-Authors: O P Ferreira, G N SilvaAbstract:In this paper we study Newton's method for solving Generalized Equations in Banach spaces. We show that under strong regularity of the Generalized Equation, the method is locally convergent to a solution with superlinear/quadratic rate. The presented analysis is based on Banach Perturbation Lemma for Generalized Equation and the classical Lipschitz condition on the derivative is relaxed by using a general majorant function, which enables obtaining the optimal convergence radius, uniqueness of solution as well as unifies earlier results pertaining to Newton's method theory.
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Convergence of the Newton's Method for Generalized Equations under the majorant condition
arXiv: Numerical Analysis, 2016Co-Authors: G N SilvaAbstract:In this paper we consider Newton's method for solving the Generalized Equation in Hilbert spaces of the type $F(x)+T(x)\ni 0$, where $F$ is a Fr\'echet differentiable function and $T$ is a set-valued and maximal monotone. We show that this method is local quadratically convergent to a solution. The analysis presented based on Banach Perturbation Lemma for Generalized Equation and the majorant condition relaxing Lipschitz continuity of the derivative $F'$, allow to obtain the optimal convergence radius, uniqueness of solution and also unify some result pertaining the Newton's method theory.
O P Ferreira - One of the best experts on this subject based on the ideXlab platform.
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Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds
Journal of Optimization Theory and Applications, 2017Co-Authors: O P Ferreira, Célia Jean-alexis, Alain PietrusAbstract:This paper is focused on the problem of finding a singularity of the sum of two vector fields defined on a Hadamard manifold, or more precisely, the study of a Generalized Equation in a Riemannian setting. We extend the concept of metric regularity to the Riemannian setting and investigate its relationship with the Generalized Equation in this new context. In particular, a version of Graves’s theorem is presented and we also define some concepts related to metric regularity, including the Aubin property and the strong metric regularity of set-valued vector fields. A conceptual method for finding a singularity of the sum of two vector fields is also considered. This method has as particular instances: the proximal point method, Newton’s method, and Zincenko’s method on Hadamard manifolds. Under the assumption of metric regularity at the singularity, we establish that the methods are well defined in a suitable neighborhood of the singularity. Moreover, we also show that each sequence generated by these methods converges to this singularity at a superlinear rate.
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kantorovich s theorem on newton s method for solving strongly regular Generalized Equation
Siam Journal on Optimization, 2017Co-Authors: O P Ferreira, G N SilvaAbstract:In this paper, we consider Newton's method for solving a Generalized Equation. We show that under strong regularity of the Equation and on the condition that the starting point satisfies Kantorovich's assumptions, the method is quadratically convergent to a solution, which is unique within a suitable neighborhood of the starting point. Our analysis, which is based on Kantorovich's majorant technique, enables us to obtain a convergence result under the Lipschitz, Smale, and Nesterov--Nemirovskii self-concordant conditions.
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kantorovich s theorem on newton s method for solving strongly regular Generalized Equation
arXiv: Numerical Analysis, 2016Co-Authors: O P Ferreira, G N SilvaAbstract:In this paper we consider the Newton's method for solving the Generalized Equation of the form $ f(x) +F(x) \ni 0, $ where $f:{\Omega}\to Y$ is a continuously differentiable mapping, $X$ and $Y$ are Banach spaces, $\Omega\subseteq X$ an open set and $F:X \rightrightarrows Y$ 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.
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Local convergence analysis of Newton's method for solving strongly regular Generalized Equations
arXiv: Numerical Analysis, 2016Co-Authors: O P Ferreira, G N SilvaAbstract:In this paper we study Newton's method for solving Generalized Equations in Banach spaces. We show that under strong regularity of the Generalized Equation, the method is locally convergent to a solution with superlinear/quadratic rate. The presented analysis is based on Banach Perturbation Lemma for Generalized Equation and the classical Lipschitz condition on the derivative is relaxed by using a general majorant function, which enables obtaining the optimal convergence radius, uniqueness of solution as well as unifies earlier results pertaining to Newton's method theory.
Asen L. Dontchev - One of the best experts on this subject based on the ideXlab platform.
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On Existence of Solutions of Parametrized Generalized Equations
Set-Valued and Variational Analysis, 2020Co-Authors: Asen L. DontchevAbstract:In this paper we study existence of solutions to the Generalized Equation 0 ∈ f ( p , x ) + F ( x ), where f is a function, F is a set-valued mapping, and p is a parameter. Conditions are given, in terms of metric regularity of F , local convex-valuedness of F ^− 1, and partial calmness of f with respect to x uniformly in p , for the property that, for any p near the reference value, the Generalized Equation has a solution at a certain distance from the reference solution. Some corollaries and applications of this result are also presented.
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Metrically Regular Differential Generalized Equations
SIAM Journal on Control and Optimization, 2018Co-Authors: Radek Cibulka, Asen L. Dontchev, Mikhail Krastanov, Vladimir M. VeliovAbstract:In this paper we consider a control system coupled with a Generalized Equation, which we call a differential Generalized Equation (DGE). This model covers a large territory in control and optimizat...
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newton s method for Generalized Equations a sequential implicit function theorem
Mathematical Programming, 2010Co-Authors: Asen L. Dontchev, R T RockafellarAbstract:In an extension of Newton’s method to Generalized Equations, we carry further the implicit function theorem paradigm and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton’s iterates as elements of a sequence space. An inverse function version of this result shows that the strong regularity of the mapping associated with the Newton sequences is equivalent to the strong regularity of the Generalized Equation mapping.
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Newton’s method for Generalized Equations: a sequential implicit function theorem
Mathematical Programming, 2009Co-Authors: Asen L. Dontchev, R T RockafellarAbstract:In an extension of Newton’s method to Generalized Equations, we carry further the implicit function theorem paradigm and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton’s iterates as elements of a sequence space. An inverse function version of this result shows that the strong regularity of the mapping associated with the Newton sequences is equivalent to the strong regularity of the Generalized Equation mapping.
R T Rockafellar - One of the best experts on this subject based on the ideXlab platform.
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newton s method for Generalized Equations a sequential implicit function theorem
Mathematical Programming, 2010Co-Authors: Asen L. Dontchev, R T RockafellarAbstract:In an extension of Newton’s method to Generalized Equations, we carry further the implicit function theorem paradigm and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton’s iterates as elements of a sequence space. An inverse function version of this result shows that the strong regularity of the mapping associated with the Newton sequences is equivalent to the strong regularity of the Generalized Equation mapping.
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Newton’s method for Generalized Equations: a sequential implicit function theorem
Mathematical Programming, 2009Co-Authors: Asen L. Dontchev, R T RockafellarAbstract:In an extension of Newton’s method to Generalized Equations, we carry further the implicit function theorem paradigm and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton’s iterates as elements of a sequence space. An inverse function version of this result shows that the strong regularity of the mapping associated with the Newton sequences is equivalent to the strong regularity of the Generalized Equation mapping.
Gerald Schubert - One of the best experts on this subject based on the ideXlab platform.
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thermal gravitational wind Equation for the wind induced gravitational signature of giant gaseous planets mathematical derivation numerical method and illustrative solutions
The Astrophysical Journal, 2015Co-Authors: Keke Zhang, Dali Kong, Gerald SchubertAbstract:The standard thermal wind Equation (TWE) relating the vertical shear of a flow to the horizontal density gradient in an atmosphere has been used to calculate the external gravitational signature produced by zonal winds in the interiors of giant gaseous planets. We show, however, that in this application the TWE needs to be Generalized to account for an associated gravitational perturbation. We refer to the Generalized Equation as the thermal-gravitational wind Equation (TGWE). The Generalized Equation represents a two-dimensional kernel integral Equation with the Green's function in its integrand and is hence much more difficult to solve than the standard TWE. We develop an extended spectral method for solving the TGWE in spherical geometry. We then apply the method to a generic gaseous Jupiter-like object with idealized zonal winds. We demonstrate that solutions of the TGWE are substantially different from those of the standard TWE. We conclude that the TGWE must be used to estimate the gravitational signature of zonal winds in giant gaseous planets.
