arXiv subjects

David Torregrosa-Belén

Publications and source records attributed to David Torregrosa-Belén.

11 recordsLinked to original sources

arXiv2026

Optimal Diagonal Preconditioning Beyond Worst-Case Conditioning: Theory and Practice of Omega Scaling

We study optimal diagonal preconditioning using the classical worst-case $κ$-condition number and the averaging-based $ω$-condition number. For the $κ$-optimal preconditioning problem, we derive an affine-based pseudoconvex reformulation with three key advantages: all stationary points are global minima, subgradients are inexpensive to compute, and the optimization variable is an $n$-dimensional vector rather than an $n\times n$ matrix as in semidefinite programming (SDP) approaches. We develop a simple and highly efficient subgradient method, with convergence guarantees, for solving this pseudoconvex formulation that is substantially more scalable and accurate than existing SDP-based methods. For the $ω$-condition number, we provide explicit characterizations of optimal diagonal and block diagonal preconditioners. In particular, we show that several classical preconditioners, including Jacobi and row/column normalization, are $ω$-optimal, and that matrix balancing schemes monotonically reduce $ω$ and converge to stationary points of the two-sided problem. To the best of our knowledge, this is the first unified and explicit characterization of optimality conditions for both $κ$ and $ω$-based preconditioning. Our numerical experiments further reveal a striking phenomenon: although $κ$-optimal preconditioners achieve stronger reductions in the worst-case condition number, $ω$-optimal preconditioners are substantially cheaper to compute and yield better performance for iterative methods such as preconditioned conjugate gradient (PCG) and least squares method (LSQR). Moreover, applying $ω$-optimal scaling to linear systems that are already $κ$-optimally preconditioned leads to further improvements in PCG iterations.

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arXiv2026

Nonmonotone subgradient methods based on a local descent lemma

In this paper we present a nonmonotone line search subgradient algorithm tailored to upper-$\mathcal{C}^2$ functions. This is a family of nonsmooth and nonconvex functions that satisfies a nonsmooth and local version of the descent lemma, making them suitable for line searches. We prove subsequential convergence of the proposed algorithm to a stationary point of the optimization problem. Our approach allows us to cover the setting of various subgradient algorithms, including Newton and quasi-Newton methods. In addition, we propose a specification of the general scheme, named Self-adaptive Nonmonotone Subgradient Method (SNSM), which automatically updates the parameters of the line search. Particular attention is paid to the minimum sum-of-squares clustering problem, for which we provide a concrete implementation of SNSM. We conclude with some numerical experiments where we exhibit the advantages of SNSM in comparison with some known algorithms.

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arXiv2026

Randomized block proximal method with locally Lipschitz continuous gradient

Block-coordinate algorithms are recognized to furnish efficient iterative schemes for addressing large-scale problems, especially when the computation of full derivatives entails substantial memory requirements and computational efforts. In this paper, we propose a randomized block proximal gradient algorithm for minimizing the sum of a smooth function and a separable proper lower semicontinuous function, both possibly nonconvex. In contrast to previous work, we only assume that the partial gradients of the smooth function are locally Lipschitz continuous with respect to their block of coordinates. At each iteration, the method adaptively selects a proximal stepsize to satisfy a sufficient decrease condition without requiring knowledge of the local Lipschitz moduli of the partial gradients of the smooth function. Hence, our work extends the applicability of randomized block-coordinate descent methods to more general problems where global Lipschitz gradient assumptions are not satisfied. In addition, our algorithm incorporates the possibility of conducting an additional boosted linesearch to enhance the performance of the algorithm. Our main result establishes subsequential convergence to a stationary point of the problem almost surely. Numerical experiments in nonnegative matrix factorization report that our method competes well with recognized algorithms. Furthermore, a symmetric variant of nonnegative matrix factorization illustrates the advantage of the boosted linesearch in problems where the proximal stepsize cannot be tightly estimated in accordance to the local Lipschitz modulus.

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arXiv2026

A proximal subgradient method for nonconvex stochastic optimization under the Kurdyka-Łojasiewicz condition

This work introduces a proximal stochastic subgradient method for minimizing the sum of an expected cost, whose integrand is potentially nonsmooth and nonconvex, and a lower semicontinuous, prox-bounded function. We target a broad class of integrands obeying a nonsmooth, localized variant of the descent lemma in the decision variable, a structural assumption that simultaneously covers smooth losses with Lipschitz gradient and differences of such losses with convex functions. At each iteration the expected cost is replaced by a sample average that is progressively refined, and the proximal-subgradient stepsize is selected by an Armijo-type line search enforcing a sufficient-decrease property up to stochastic errors induced by the sample-based approximation. This framework accommodates substantially more general problem formulations than existing methods, in particular, it requires neither (weak) convexity of the regularizer nor a uniform bound on the variance of the stochastic oracle, and our analysis yields convergence guarantees that are new even in the smooth setting. Specifically, we establish almost sure convergence of the sequence of function values and stationarity of every accumulation point of the trajectories under the relaxed requirement that the sample-size sequence be merely nondecreasing and unbounded, with no prescribed growth rate. Leveraging the Kurdyka-Lojasiewicz (KL) property, we further upgrade this subsequential guarantee to convergence of the whole trajectory to a single stationary point. Finally, for exponential-type KL desingularizing functions and polynomially growing sample sizes, we derive explicit polynomial convergence rates, up to a logarithmic factor, for both the function values and the iterates.

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arXiv2026

Projection, Degeneracy, and Singularity Degree for Spectrahedra

Facial reduction, FR, is a regularization technique for convex programs where the strict feasibility constraint qualification, CQ, fails.Though this CQ holds generically, failure is pervasive in applications such as semidefinite relaxations of hard discrete optimization problems. In this paper we relate FR to the analysis of the convergence behaviour of a semismooth Newton root finding method for the projection onto a spectrahedron, i.e., onto the intersection of a linear manifold and the semidefinite cone. We examine the effect of failure of strict feasibility on the projection problem. In the process, we derive an elegant formula for the projection onto a face of the semidefinite cone obtained via regularization and discuss pathologies that arise in the absence of strict feasibility. We show further that the ill-conditioning of the Jacobian of the Newton method near optimality characterizes the degeneracy of the nearest point in the spectrahedron. We apply the results, both theoretically and empirically, to the problem of finding nearest points to the sets of: (i) correlation matrices or the elliptope; and (ii) semidefinite relaxations of permutation matrices or the vontope, i.e., the feasible sets for the semidefinite relaxations of the max-cut and quadratic assignment problems, respectively.

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arXiv2024

The $ω$-Condition Number: Applications to Optimal Preconditioning and Low Rank Generalized Jacobian Updating

Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g.,in quasi-Newton methods. Motivated by the latter, we study a nonclassic matrix condition number, the $ω$-condition number, $ω$ for short. $ω$ is the ratio of the arithmetic and geometric means of the singular values, rather than largest and smallest. Moreover, unlike the latter classical $κ$ condition number, $ω$ is not invariant under inversion, an important point that allows one to recall that it is the conditioning of the inverse that is important. Our study is in the context of optimal conditioning for: (i) low rank updating of generalized Jacobians arising in the context of nonsmooth Newton methods; and (ii) iterative methods for linear systems; (iia) clustering of eigenvalues; (iib) convergence rates; and (iic) estimating the actual condition of a linear system. We emphasize that the simple functions in $ω$ allow one to exploit optimality conditions and derive explicit formulae for $ω$-optimal preconditioners of special structure. Connections to partial Cholesky type sparse preconditioners are made that modify the iterates of Cholesky decomposition by including the entire diagonal at each iteration. Our results confirm the efficacy of using the $ω$-condition number compared to the classical $κ$-condition number.

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arXiv2024

The superiorization method with restarted perturbations for split minimization problems with an application to radiotherapy treatment planning

In this paper we study the split minimization problem that consists of two constrained minimization problems in two separate spaces that are connected via a linear operator that maps one space into the other. To handle the data of such a problem we develop a superiorization approach that can reach a feasible point with reduced (not necessarily minimal) objective function values. The superiorization methodology is based on interlacing the iterative steps of two separate and independent iterative processes by perturbing the iterates of one process according to the steps dictated by the other process. We include in our developed method two novel elements. The first one is the permission to restart the perturbations in the superiorized algorithm which results in a significant acceleration and increases the computational efficiency. The second element is the ability to independently superiorize subvectors. This caters to the needs of real-world applications, as demonstrated here for a problem in intensity-modulated radiation therapy treatment planning.

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arXiv2023

The Boosted Double-Proximal Subgradient Algorithm for Nonconvex Optimization

In this paper we introduce the Boosted Double-proximal Subgradient Algorithm (BDSA), a novel splitting algorithm designed to address general structured nonsmooth and nonconvex mathematical programs expressed as sums and differences of composite functions. BDSA exploits the combined nature of subgradients from the data and proximal steps, and integrates a line-search procedure to enhance its performance. While BDSA encompasses existing schemes proposed in the literature, it extends its applicability to more diverse problem domains. We establish the convergence of BDSA under the Kurdyka--Lojasiewicz property and provide an analysis of its convergence rate. To evaluate the effectiveness of BDSA, we introduce a novel family of challenging test functions with an abundance of critical points. We conduct comparative evaluations demonstrating its ability to effectively escape non-optimal critical points. Additionally, we present two practical applications of BDSA for testing its efficacy, namely, a constrained minimum-sum-of-squares clustering problem and a nonconvex generalization of Heron's problem.

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arXiv2022

Distributed Forward-Backward Methods for Ring Networks

In this work, we propose and analyse forward-backward-type algorithms for finding a zero of the sum of finitely many monotone operators, which are not based on reduction to a two operator inclusion in the product space. Each iteration of the studied algorithms requires one resolvent evaluation per set-valued operator, one forward evaluation per cocoercive operator, and two forward evaluations per monotone operator. Unlike existing methods, the structure of the proposed algorithms are suitable for distributed, decentralised implementation in ring networks without needing global summation to enforce consensus between nodes.

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arXiv2022

A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed with linear operators. Our main contribution is establishing the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Specifically, the proposed scheme reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators. We prove the convergence of this new algorithm and analyze its performance in a problem arising in image deblurring and denoising. This work also contributes to the theory of resolvent splitting algorithms by extending the minimal lifting theorem recently proved by Malitsky and Tam to schemes with resolvent parameters.

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arXiv2022

A direct proof of convergence of Davis-Yin splitting algorithm allowing larger stepsizes

This note is devoted to the splitting algorithm proposed by Davis and Yin in 2017 for computing a zero of the sum of three maximally monotone operators, with one of them being cocoercive. We provide a direct proof that guarantees its convergence when the stepsizes are smaller than four times the cocoercivity constant, thus doubling the size of the interval established by Davis and Yin. As a by-product, the same conclusion applies to the forward-backward splitting algorithm. Further, we use the notion of "strengthening" of a set-valued operator to derive a new splitting algorithm for computing the resolvent of the sum. Last but not least, we provide some numerical experiments illustrating the importance of appropriately choosing the stepsize and relaxation parameters of the algorithms.

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