arXiv subjects

Geovani N. Grapiglia

Publications and source records attributed to Geovani N. Grapiglia.

7 recordsLinked to original sources

arXiv2026

Robust Sparse Identification of Nonlinear Dynamics via Least Trimmed Squares

In this work, we propose a robust Sparse Identification of Nonlinear Dynamics (SINDy) pipeline for handling datasets corrupted by noise and outliers. The method decouples outlier filtering from sparse regression by combining Iterative Least Trimmed Squares (ILTS) with Sequentially Thresholded Least Squares (STLS). Unlike standard approaches that treat all observations uniformly within a single regression stage, the proposed ILTS-SINDy framework first applies an ILTS procedure that iteratively minimizes the sum of the smallest squared residuals to identify the most reliable observations without prior knowledge of outliers, after which STLS is used to recover a parsimonious governing model. Extensive numerical experiments show that ILTS-SINDy can significantly outperform existing robust SINDy variants across a range of outlier contamination levels, with performance maintained even under settings with up to $20\%$ corrupted observations.

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arXiv2026

Complexity of quadratic penalty methods with adaptive accuracy under a PL condition for the constraints

We study the quadratic penalty method (QPM) for smooth nonconvex optimization problems with equality constraints. Assuming the constraint violation satisfies the PL condition near the feasible set, we derive sharper worst-case complexity bounds for obtaining approximate first-order KKT points. When the objective and constraints are twice continuously differentiable, we show that QPM equipped with a suitable first-order inner solver requires at most $O(\varepsilon_{0}^{-1}\varepsilon_{1}^{-2})$ first-order oracle calls to find an $(\varepsilon_{0},\varepsilon_{1})$-approximate KKT point -- that is, a point that is $\varepsilon_{0}$-approximately feasible and $\varepsilon_{1}$-approximately stationary. Furthermore, when the objective and constraints are three times continuously differentiable, we show that QPM with a suitable second-order inner solver requires at most $O\left(\varepsilon_{0}^{-1/2}\varepsilon_{1}^{-3/2}\right)$ second-order oracle calls to find an $(\varepsilon_{0},\varepsilon_{1})$-approximate KKT point. We also introduce an adaptive, feasibility-aware stopping criterion for the subproblems, which relaxes the stationarity tolerance when far from feasibility. This rule preserves all theoretical guarantees while substantially reducing computational effort in practice.

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arXiv2025

A Riemannian AdaGrad-Norm Method

We propose a manifold AdaGrad-Norm method (\textsc{MAdaGrad}), which extends the norm version of AdaGrad (AdaGrad-Norm) to Riemannian optimization. In contrast to line-search schemes, which may require several exponential map computations per iteration, \textsc{MAdaGrad} requires only one. Assuming the objective function $f$ has Lipschitz continuous Riemannian gradient, we show that the method requires at most $\mathcal{O}(\varepsilon^{-2})$ iterations to compute a point $x$ such that $\|\operatorname{grad} f(x)\|\leq \varepsilon$. Under the additional assumptions that $f$ is geodesically convex and the manifold has sectional curvature bounded from below, we show that the method takes at most $\mathcal{O}(\varepsilon^{-1})$ to find $x$ such that $f(x)-f_{low}\leqε$, where $f_{low}$ is the optimal value. Moreover, if $f$ satisfies the Polyak--Łojasiewicz condition globally on the manifold, we establish a complexity bound of $\mathcal{O}(\log(\varepsilon^{-1}))$, provided that the norm of the initial Riemannian gradient is sufficiently large. For the manifold of symmetric positive definite matrices, we construct a family of nonconvex functions satisfying the PL condition. Numerical experiments illustrate the remarkable performance of \textsc{MAdaGrad} in comparison with Riemannian Steepest Descent equipped with Armijo line-search.

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arXiv2025

Active-set Newton-MR methods for nonconvex optimization problems with bound constraints

This paper presents active-set methods for minimizing nonconvex twice-continuously differentiable functions subject to bound constraints. Within the faces of the feasible set, we employ descent methods with Armijo line search, utilizing approximated Newton directions obtained through the Minimum Residual (MINRES) method. To escape the faces, we investigate the use of the Spectral Projected Gradient (SPG) method and a tailored variant of the Cubic Regularization of Newton's method for bound-constrained problems. We provide theoretical guarantees, demonstrating that when the objective function has a Lipschitz continuous gradient, the SPG-based method requires no more than $\mathcal{O}(nε^{-2})$ oracle calls to find $ε$-approximate stationary points, where $n$ is the problem dimension. Furthermore, if the objective function also has a Lipschitz continuous Hessian, we show that the method based on cubic regularization requires no more than $\mathcal{O}\left(n|\log_{2}(ε)|ε^{-3/2}\right)$ oracle calls to achieve the same goal. We emphasize that, under certain hypotheses, the method achieves $O(ε^{3/2})$ descent within the faces without resorting to cubic regularization. Numerical experiments are conducted to compare the proposed methods with existing active-set methods, highlighting the potential benefits of using MINRES instead of the Conjugate Gradient (CG) method for approximating Newton directions.

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arXiv2025

Sub-sampled Trust-Region Methods with Deterministic Worst-Case Complexity Guarantees

In this paper, we develop and analyze sub-sampled trust-region methods for solving finite-sum optimization problems. These methods employ subsampling strategies to approximate the gradient and Hessian of the objective function, significantly reducing the overall computational cost. We propose a novel adaptive procedure for deterministically adjusting the sample size used for gradient (or gradient and Hessian) approximations. Furthermore, we establish worst-case iteration complexity bounds for obtaining approximate stationary points. More specifically, for a given $\varepsilon_g, \varepsilon_H\in (0,1)$, it is shown that an $\varepsilon_g$-approximate first-order stationary point is reached in at most $\mathcal{O}({\varepsilon_g}^{-2} )$ iterations, whereas an $(\varepsilon_g,\varepsilon_H)$-approximate second-order stationary point is reached in at most $\mathcal{O}(\max\{\varepsilon_{g}^{-2}\varepsilon_{H}^{-1},\varepsilon_{H}^{-3}\})$ iterations. Finally, numerical experiments illustrate the effectiveness of our new subsampling technique.

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arXiv2021

A Generalized Worst-Case Complexity Analysis for Non-Monotone Line Searches

We study the worst-case complexity of a non-monotone line search framework that covers a wide variety of known techniques published in the literature. In this framework, the non-monotonicity is controlled by a sequence of nonnegative parameters. We obtain complexity bounds to achieve approximate first-order optimality even when this sequence is not summable.

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arXiv2021

On the Complexity of an Augmented Lagrangian Method for Nonconvex Optimization

In this paper we study the worst-case complexity of an inexact Augmented Lagrangian method for nonconvex constrained problems. Assuming that the penalty parameters are bounded, we prove a complexity bound of $\mathcal{O}(|\log(ε)|)$ outer iterations for the referred algorithm to generate an $ε$-approximate KKT point, for $ε\in (0,1)$. When the penalty parameters are unbounded, we prove an outer iteration complexity bound of $\mathcal{O}\left(ε^{-2/(α-1)}\right)$, where $α>1$ controls the rate of increase of the penalty parameters. For linearly constrained problems, these bounds yield to evaluation complexity bounds of $\mathcal{O}(|\log(ε)|^{2}ε^{-2})$ and $\mathcal{O}\left(ε^{-\left(\frac{2(2+α)}{α-1}+2\right)}\right)$, respectively, when appropriate first-order methods ($p=1$) are used to approximately solve the unconstrained subproblems at each iteration. In the case of problems having only linear equality constraints, the latter bounds are improved to $\mathcal{O}(|\log(ε)|^{2}ε^{-(p+1)/p})$ and $\mathcal{O}\left(ε^{-\left(\frac{4}{α-1}+\frac{p+1}{p}\right)}\right)$, respectively, when appropriate $p$-order methods ($p\geq 2$) are used as inner solvers.

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