arXiv subjects

Gilson N. Silva

Publications and source records attributed to Gilson N. Silva.

7 recordsLinked to original sources

arXiv2026

An Adaptive Proximal ADMM for Nonconvex Linearly Constrained Composite Programs

This paper develops an adaptive proximal alternating direction method of multipliers (ADMM) for solving linearly constrained, composite optimization problems under the assumption that the smooth component of the objective is weakly convex, while the non-smooth component is convex and block-separable. The proposed method is adaptive to all problem parameters, including smoothness and weak convexity constants, and allows each of its block proximal subproblems to be inexactly solved. Each iteration of our adaptive proximal ADMM consists of two steps: the sequential solution of each block proximal subproblem; and adaptive tests to decide whether to perform a full Lagrange multiplier and/or penalty parameter update(s). Without any rank assumptions on the constraint matrices, it is shown that the adaptive proximal ADMM obtains an approximate first-order stationary point of the constrained problem in a number of iterations that matches the state-of-the-art complexity for the class of proximal ADMM's. The three proof-of-concept numerical experiments that conclude the paper suggest our adaptive proximal ADMM enjoys significant computational benefits.

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arXiv2026

A Regularized Hessian-Free Inexact Newton-Type Method with Global $\mathcal{O}(k^{-2})$ Convergence

We propose a regularized Hessian-free Newton-type method for minimizing smooth convex functions with Lipschitz continuous Hessians. The algorithm constructs an approximate Hessian by finite differences and selects the regularization parameter through an adaptive criterion that ensures sufficient decrease and gradient control. We prove that the method achieves a global $\mathcal{O}(k^{-2})$ convergence rate, matching the best known bound for second-order methods. A modified variant incorporating the exact Hessian when available enjoys local quadratic convergence under standard assumptions. Despite its simplicity, this variant is computationally faster than the \emph{Regularized Newton Method} of Mishchenko (2023) across several convex benchmark problems. Our analysis also provides explicit bounds on the regularization sequence and a worst-case iteration complexity of order $\mathcal{O}(\varepsilon^{-2})$. The proposed framework thus unifies regularized and Hessian-free Newton-type schemes, offering a theoretically sound and practically efficient alternative for smooth convex optimization.

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arXiv2025

An Adaptive Proximal Point Method for Nonsmooth and Nonconvex Optimization on Hadamard Manifolds

This paper addresses a class of nonsmooth and nonconvex optimization problems defined on complete Riemannian manifolds. The objective function has a composite structure, combining convex, differentiable, and lower semicontinuous terms, thereby generalizing the classical framework of difference-of-convex programming. Motivated by recent advances in proximal point methods in Euclidean and Riemannian settings, we propose two variants: one that uses the Lipschitz constant of the gradient of the smooth part, suitable when this parameter is accessible, and another that dispenses with such knowledge, expanding its applicability. We analyze the complexity of both approaches, establish their convergence, and illustrate their effectiveness through numerical experiments.

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arXiv2024

Inexact Newton Methods for Solving Generalized Equations on Riemannian Manifolds

The convergence of inexact Newton methods is studied for solving generalized equations on Riemannian manifolds by using the metric regularity property, which is also explored. Under appropriate conditions and without any additional geometric assumptions, local convergence results with linear and quadratic rates, as well as a semi-local convergence result, are obtained for the proposed method. Finally, the theory is applied to the problem of finding a singularity for the sum of two vector fields. In particular, the KKT system for the constrained Riemannian center of mass on the sphere is explored numerically.

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arXiv2024

An Adaptive Cubic Regularization quasi-Newton Method on Riemannian Manifolds

A quasi-Newton method with cubic regularization is designed for solving Riemannian unconstrained nonconvex optimization problems. The proposed algorithm is fully adaptive with at most ${\cal O} (ε_g^{-3/2})$ iterations to achieve a gradient smaller than $ε_g$ for given $ε_g$, and at most $\mathcal O(\max\{ ε_g^{-\frac{3}{2}}, ε_H^{-3} \})$ iterations to reach a second-order stationary point respectively. Notably, the proposed algorithm remains applicable even in cases of the gradient and Hessian of the objective function unknown. Numerical experiments are performed with gradient and Hessian being approximated by forward finite-differences to illustrate the theoretical results and numerical comparison.

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arXiv2016

Local convergence of Newton's method for solving generalized equations with monotone operator

In this paper we study Newton's method for solving the generalized equation $F(x)+T(x)\ni 0$ in Hilbert spaces, where $F$ is a Fréchet differentiable function and $T$ is set-valued and maximal monotone. We show that this method is local quadratically convergent to a solution. Using the idea of majorant condition on the nonlinear function which is associated to the generalized equation, the convergence of the method, the optimal convergence radius and results on the convergence rate are established. The advantage of working with a majorant condition rests in the fact that it allow to unify several convergence results pertaining to Newton's method.

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arXiv2016

On the Kantorovich's theorem for Newton's method for solving generalized equations under the majorant condition

In this paper we consider a version of the Kantorovich's theorem for solving the generalized equation $F(x)+T(x)\ni 0$, where $F$ is a Fréchet derivative function and $T$ is a set-valued and maximal monotone acting between Hilbert spaces. We show that this method is quadratically convergent to a solution of $F(x)+T(x)\ni 0$. We have used the idea of majorant function, which relaxes the Lipschitz continuity of the derivative $F'$. It allows us to obtain the optimal convergence radius, uniqueness of solution and also to solving generalized equations under Smale's condition.

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