arXiv subjects

Paulo J. S. Silva

Publications and source records attributed to Paulo J. S. Silva.

5 recordsLinked to original sources

arXiv2026

Parallel Newton methods for the continuous quadratic knapsack problem: A Jacobi and Gauss-Seidel tale

The continuous quadratic knapsack (CQK) problem involves minimizing a diagonal convex quadratic function subject to box constraints and a single linear equality constraint. It has numerous applications in resource allocation, multicommodity flow, machine learning, and classical optimization tasks such as Lagrangian relaxation and quasi-Newton updates. In this work, we revisit the semismooth Newton method introduced by Cominetti, Mascarenhas, and Silva. We demonstrate that the method can be significantly improved in two directions. First, for projections onto the simplex or the $\ell_1$-ball, it can incorporate Condat's highly effective initial multiplier guess. Second, it can serve as a flexible foundation for CQK algorithms, allowing for different parallel variants tailored to exploit CPU and GPU computational models. These improvements are implemented in the open-source Julia package \texttt{NewtonCQK.jl}. We present extensive numerical tests comparing this implementation with other state-of-the-art solvers, demonstrating its superior efficiency and scalability.

math.OC↗

arXiv2026

Basis pursuit by inconsistent alternating projections

Basis pursuit is the problem of finding a vector with smallest $\ell_1$-norm among the solutions of a given linear system of equations. It is a well-known convex relaxation of the sparse affine feasibility problem, where sparse solutions to underdetermined systems are sought. Since basis pursuit admits a linear programming reformulation, standard LP solvers are directly applicable. We instead address the basis pursuit directly in its $\ell_1$-minimization form, without LP reformulation, via a scheme that uses alternating projections in its subproblems. These subproblems are designed to be inconsistent in the sense that they relate to two non-intersecting sets. Recently in [R. Behling, Y. Bello-Cruz and L.-R. Santos, SIAM J. Optim., 31 (2021), pp. 2863-2892], inconsistency coming from infeasibility has been shown to accelerate convergence of alternating projections. We deliberately enforce this inconsistency by constructing subproblems whose feasible sets are disjoint by design. We prove that the resulting $\ell_1$-radii converge linearly to the optimal value, and that when the solution is unique, all generated sequences converge linearly to it at a rate governed by a natural error bound between the feasible set and the optimal $\ell_1$-ball. The proposed method is numerically competitive against state-of-the-art open-source solvers on synthetic and real-world instances.

math.OC↗

arXiv2025

Parallel block coordinate descent methods with identification strategies

This work presents a parallel variant of the algorithm introduced in [Acceleration of block coordinate descent methods with identification strategies Comput. Optim. Appl. 72(3):609--640, 2019] to minimize the sum of a partially separable smooth convex function and a possibly non-smooth block-separable convex function under simple constraints. It achieves better efficiency by using a strategy to identify the nonzero coordinates that allows the computational effort to be focused on using a nonuniform probability distribution in the selection of the blocks. Parallelization is achieved by extending the theoretical results from Richtárik and Takáč [Parallel coordinate descent methods for big data optimization, Math. Prog. Ser. A 156:433--484, 2016]. We present convergence results and comparative numerical experiments on regularized regression problems using both synthetic and real data.

math.OC↗

arXiv2021

Smart testing and critical care bed sharing for COVID-19 control

During the early months of the current COVID-19 pandemic, social-distancing measures effectively slowed disease transmission in many countries in Europe and Asia, but the same benefits have not been observed in some developing countries such as Brazil. In part, this is due to a failure to organise systematic testing campaigns at nationwide or even regional levels. To gain effective control of the pandemic, decision-makers in developing countries, particularly those with large populations, must overcome difficulties posed by an unequal distribution of wealth combined with low daily testing capacities. The economic infrastructure of the country, often concentrated in a few cities, forces workers to travel from commuter cities and rural areas, which induces strong nonlinear effects on disease transmission. In the present study, we develop a smart testing strategy to identify geographic regions where COVID-19 testing could most effectively be deployed to limit further disease transmission. The strategy uses readily available anonymised mobility and demographic data integrated with intensive care unit (ICU) occupancy data and city-specific social-distancing measures. Taking into account the heterogeneity of ICU bed occupancy in differing regions and the stages of disease evolution, we use a data-driven study of the Brazilian state of Sao Paulo as an example to show that smart testing strategies can rapidly limit transmission while reducing the need for social-distancing measures, thus returning life to a so-called new normal, even when testing capacity is limited.

physics.soc-ph↗

arXiv2020

TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data

Multidimensional Projection is a fundamental tool for high-dimensional data analytics and visualization. With very few exceptions, projection techniques are designed to map data from a high-dimensional space to a visual space so as to preserve some dissimilarity (similarity) measure, such as the Euclidean distance for example. In fact, although adopting distinct mathematical formulations designed to favor different aspects of the data, most multidimensional projection methods strive to preserve dissimilarity measures that encapsulate geometric properties such as distances or the proximity relation between data objects. However, geometric relations are not the only interesting property to be preserved in a projection. For instance, the analysis of particular structures such as clusters and outliers could be more reliably performed if the mapping process gives some guarantee as to topological invariants such as connected components and loops. This paper introduces TopoMap, a novel projection technique which provides topological guarantees during the mapping process. In particular, the proposed method performs the mapping from a high-dimensional space to a visual space, while preserving the 0-dimensional persistence diagram of the Rips filtration of the high-dimensional data, ensuring that the filtrations generate the same connected components when applied to the original as well as projected data. The presented case studies show that the topological guarantee provided by TopoMap not only brings confidence to the visual analytic process but also can be used to assist in the assessment of other projection methods.

cs.GR↗