arXiv subjects

Marcos Raydan

Publications and source records attributed to Marcos Raydan.

6 recordsLinked to original sources

arXiv2026

Derivative-free optimization approach for structured symmetric matrices with fixed eigenvalues

A Derivative-Free Optimization (DFO) model is developed and analyzed for solving inverse structured symmetric matrix problems for which the eigenvalues are specified. Some (zero and nonzero) entries are preassigned and cannot be changed, while others should be nonzero but their values are not given. The rest of the entries are completely free. The obtained matrix must meet these criteria and have the specified eigenvalues. This specialized inverse eigenvalue problem is relevant to various applications and is linked to determining the graph, with weights on the undirected edges, of the matrix associated with its sparse pattern. Our novel optimization model requires computing the eigenvalues of a symmetric matrix to evaluate the non-differentiable objective function. We apply deterministic DFO schemes, specifically the global variant GLODS of the well-known family of directional direct search (DDS) methods. We discuss its convergence properties which are based on the fact that the objective function of our model is Lipschitz continuous. Additionally, we explore the potential benefits of using several well-established heuristic strategies to solve the proposed optimization model. We present preliminary numerical results to illustrate and compare the performance of the considered deterministic and heuristic DFO options in various possible scenarios.

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arXiv2026

Eigenvector-based acceleration strategies for gradient-type methods

Several strategies are described and analyzed to speed-up gradient-type methods when applied to the minimization of strictly convex quadratics and strictly convex functions. The proposed techniques focus on relaxing the traditional optimal step length associated with gradient methods, including the steepest descent (SD) and the minimal residual (MR) methods. Such a relaxation avoids the well-known negative zigzag effect and allows the iterates to move in the entire space which in turn implies that every so often the search direction approaches some eigenvector of the underlying Hessian matrix. The proposed speedups then rely on taking advantage of the properties of the Lanczos method once a search direction that approaches an eigenvector has been identified in order to accelerate the convergence towards the global minimizer. After analyzing the proposed strategies, we illustrate them on the global minimization of strictly convex functions.

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arXiv2023

SLiSeS: Subsampled Line Search Spectral Gradient Method for Finite Sums

The spectral gradient method is known to be a powerful low-cost tool for solving large-scale optimization problems. In this paper, our goal is to exploit its advantages in the stochastic optimization framework, especially in the case of mini-batch subsampling that is often used in big data settings. To allow the spectral coefficient to properly explore the underlying approximate Hessian spectrum, we keep the same subsample for several iterations before subsampling again. We analyze the required algorithmic features and the conditions for almost sure convergence, and present initial numerical results that show the advantages of the proposed method.

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arXiv2022

A low-cost alternating projection approach for a continuous formulation of convex and cardinality constrained optimization

We consider convex constrained optimization problems that also include a cardinality constraint. In general, optimization problems with cardinality constraints are difficult mathematical programs which are usually solved by global techniques from discrete optimization. We assume that the region defined by the convex constraints can be written as the intersection of a finite collection of convex sets, such that it is easy and inexpensive to project onto each one of them (e.g., boxes, hyper-planes, or half-spaces). Taking advantage of a recently developed continuous reformulation that relaxes the cardinality constraint, we propose a specialized penalty gradient projection scheme combined with alternating projection ideas to solve these problems. To illustrate the combined scheme, we focus on the standard mean-variance portfolio optimization problem for which we can only invest in a preestablished limited number of assets. For these portfolio problems with cardinality constraints we present a numerical study on a variety of data sets involving real-world capital market indices from major stock markets. On those data sets we illustrate the computational performance of the proposed scheme to produce the effective frontiers for different values of the limited number of allowed assets.

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arXiv2019

Geometrical inverse matrix approximation for least-squares problems and acceleration strategies

We extend the geometrical inverse approximation approach for solving linear least-squares problems. For that we focus on the minimization of $1-\cos(X(A^TA),I)$, where $A$ is a given rectangular coefficient matrix and $X$ is the approximate inverse. In particular, we adapt the recently published simplified gradient-type iterative scheme MinCos to the least-squares scenario. In addition, we combine the generated convergent sequence of matrices with well-known acceleration strategies based on recently developed matrix extrapolation methods, and also with some deterministic and heuristic acceleration schemes which are based on affecting, in a convenient way, the steplength at each iteration. A set of numerical experiments, including large-scale problems, are presented to illustrate the performance of the different accelerations strategies.

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arXiv2015

Geometrical inverse preconditioning for symmetric positive definite matrices

We focus on inverse preconditioners based on minimizing $F(X) = 1-\cos(XA,I)$, where $XA$ is the preconditioned matrix and $A$ is symmetric and positive definite. We present and analyze gradient-type methods to minimize $F(X)$ on a suitable compact set. For that we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of $F(X)$ on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included.

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