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Behling, R., Lara, H., & Oviedo, H. (2023). Computing the completely positive factorization via alternating minimization. Numer. Linear Algebra Appl., Early Access.
Abstract: In this article, we propose a novel alternating minimization scheme for finding completely positive factorizations. In each iteration, our method splits the original factorization problem into two optimization subproblems, the first one being an orthogonal procrustes problem, which is taken over the orthogonal group, and the second one over the set of entrywise positive matrices. We present both a convergence analysis of the method and favorable numerical results.
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Oviedo, H. (2023). Proximal Point Algorithm with Euclidean Distance on the Stiefel Manifold. Mathematics, 11(11), 2414.
Abstract: In this paper, we consider the problem of minimizing a continuously differentiable function on the Stiefel manifold. To solve this problem, we develop a geodesic-free proximal point algorithm equipped with Euclidean distance that does not require use of the Riemannian metric. The proposed method can be regarded as an iterative fixed-point method that repeatedly applies a proximal operator to an initial point. In addition, we establish the global convergence of the new approach without any restrictive assumption. Numerical experiments on linear eigenvalue problems and the minimization of sums of heterogeneous quadratic functions show that the developed algorithm is competitive with some procedures existing in the literature.
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Oviedo, H., & Herrera, R. (2023). A collection of efficient retractions for the symplectic Stiefel manifold. Comput. Appl. Math., 42(4), 164.
Abstract: This article introduces a new map on the symplectic Stiefel manifold. The operation that requires the highest computational cost to compute the novel retraction is a inversion of size 2p-by-2p, which is much less expensive than those required for the available retractions in the literature. Later, with the new retraction, we design a constraint preserving gradient method to minimize smooth functions defined on the symplectic Stiefel manifold. To improve the numerical performance of our approach, we use the non-monotone line-search of Zhang and Hager with an adaptive Barzilai-Borwein type step-size. Our numerical studies show that the proposed procedure is computationally promising and is a very good alternative to solve large-scale optimization problems over the symplectic Stiefel manifold.
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