arXiv subjects

G. N. Silva

Publications and source records attributed to G. N. Silva.

5 recordsLinked to original sources

arXiv2025

Broyden quasi-Newton secant-type method for solving constrained mixed generalized equations

This paper presents a novel variant of the Broyden quasi-Newton secant-type method aimed at solving constrained mixed generalized equations, which can include functions that are not necessarily differentiable. The proposed method integrates the classical secant approach with techniques inspired by the Conditional Gradient method to handle constraints effectively. We establish local convergence results by applying the contraction mapping principle. Specifically, under assumptions of Lipschitz continuity, a modified Broyden update for derivative approximation, and the metric regularity property, we show that the algorithm generates a well-defined sequence that converges locally at a Q-linear rate.

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arXiv2021

On the value function for nonautonomous optimal control problems with infinite horizon

In this paper we consider nonautonomous optimal control problems of infinite horizon type, whose control actions are given by $L^1$-functions. We verify that the value function is locally Lipschitz. The equivalence between dynamic programming inequalities and Hamilton-Jacobi-Bellman (HJB) inequalities for proximal sub (super) gradients is proven. Using this result we show that the value function is a Dini solution of the HJB equation. We obtain a verification result for the class of Dini sub-solutions of the HJB equation and also prove a minimax property of the value function with respect to the sets of Dini semi-solutions of the HJB equation. We introduce the concept of viscosity solutions of the HJB equation in infinite horizon and prove the equivalence between this and the concept of Dini solutions. In the appendix we provide an existence theorem.

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arXiv2016

Local convergence analysis of Newton's method for solving strongly regular generalized equations

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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arXiv2016

Inexact Newton's method to nonlinear functions with values in a cone

The problem of finding a solution of nonlinear inclusion problems in Banach space is considered in this paper. Using convex optimization techniques introduced by Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), a robust convergence theorem for inexact Newton's method is proved. As an application, an affine invariant version of Kantorovich's theorem and Smale's α-theorem for inexact Newton's method is obtained.

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arXiv2016

Kantorovich's theorem on Newton's method for solving strongly regular generalized equation

In this paper we consider the Newton's method for solving the generalized equation of the form $ f(x) +F(x) \ni 0, $ where $f:Ω\to Y$ is a continuously differentiable mapping, $X$ and $Y$ are Banach spaces, $Ω\subseteq X$ 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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