arXiv subjects

Leandro Farias Maia

Publications and source records attributed to Leandro Farias Maia.

6 recordsLinked to original sources

arXiv2026

Convergence Analysis of an Inexact Proximal Trust-Region Method

We analyze an inexact, proximal trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function. Our algorithm leverages the proximal gradient as a benchmark to guarantee global convergence. However, evaluating the proximal gradient involves solving a nonsmooth convex optimization problem that in many applications cannot be performed exactly. As such, we develop a framework that permits inexact evaluations of the proximity operator and we show the proposed inexactness conditions yield descent properties analogous to the exact proximal gradient. We further discuss practical procedures for computing approximate proximity operators for two common applications: (i) the nonsmooth term is the sum of nonsmooth convex functions where one is composed with an affine map and (ii) using alternative inner products to facilitate the evaluation of the proximity operator. We demonstrate our algorithm and confirm our analysis with two numerical experiments arising in PDE-constrained optimization.

math.OC↗

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.

math.OC↗

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.

math.OC↗

arXiv2026

Block Decomposable Methods for Large-Scale Optimization Problems

This dissertation explores block decomposable methods for large-scale optimization problems. It focuses on alternating direction method of multipliers (ADMM) schemes and block coordinate descent (BCD) methods. Specifically, it introduces a new proximal ADMM algorithm and proposes two BCD methods. The first part of the research presents a new proximal ADMM algorithm. This method is adaptive to all problem parameters and solves the proximal augmented Lagrangian (AL) subproblem inexactly. This adaptiveness facilitates the highly efficient application of the algorithm to a broad swath of practical problems. The inexact solution of the proximal AL subproblem overcomes many key challenges in the practical applications of ADMM. The resultant algorithm obtains an approximate solution of an optimization problem in a number of iterations that matches the state-of-the-art complexity for the class of proximal ADMM schemes. The second part of the research focuses on an inexact proximal mapping for the class of block proximal gradient methods. Key properties of this operator is established, facilitating the derivation of convergence rates for the proposed algorithm. Under two error decreases conditions, the algorithm matches the convergence rate of its exactly computed counterpart. Numerical results demonstrate the superior performance of the algorithm under a dynamic error regime over a fixed one. The dissertation concludes by providing convergence guarantees for the randomized BCD method applied to a broad class of functions, known as Hölder smooth functions. Convergence rates are derived for non-convex, convex, and strongly convex functions. These convergence rates match those furnished in the existing literature for the Lipschtiz smooth setting.

math.OC↗

arXiv2026

An Inexact Weighted Proximal Trust-Region Method

In [R. J. Baraldi and D. P. Kouri, Math. Program., 201:1 (2023), pp. 559-598], the authors introduced a trust-region method for minimizing the sum of a smooth nonconvex and a nonsmooth convex function, the latter of which has an analytical proximity operator. While many functions satisfy this criterion, e.g., the $\ell_1$-norm defined on $\ell_2$, many others are precluded by either the topology or the nature of the nonsmooth term. Using the $δ$-Fréchet subdifferential, we extend the definition of the inexact proximity operator and enable its use within the aforementioned trust-region algorithm. Moreover, we augment the analysis for the standard trust-region convergence theory to handle proximity operator inexactness with weighted inner products. We first introduce an algorithm to generate a point in the inexact proximity operator and then apply the algorithm within the trust-region method to solve an optimal control problem constrained by Burgers' equation.

math.OC↗

arXiv2024

The Randomized Block Coordinate Descent Method in the Hölder Smooth Setting

This work provides the first convergence analysis for the Randomized Block Coordinate Descent method for minimizing a function that is both Hölder smooth and block Hölder smooth. Our analysis applies to objective functions that are non-convex, convex, and strongly convex. For non-convex functions, we show that the expected gradient norm reduces at an $O\left(k^{\fracγ{1+γ}}\right)$ rate, where $k$ is the iteration count and $γ$ is the Hölder exponent. For convex functions, we show that the expected suboptimality gap reduces at the rate $O\left(k^{-γ}\right)$. In the strongly convex setting, we show this rate for the expected suboptimality gap improves to $O\left(k^{-\frac{2γ}{1-γ}}\right)$ when $γ>1$ and to a linear rate when $γ=1$. Notably, these new convergence rates coincide with those furnished in the existing literature for the Lipschitz smooth setting.

math.OC↗