arXiv subjects

Jose Yunier Bello Cruz

Publications and source records attributed to Jose Yunier Bello Cruz.

5 recordsLinked to original sources

arXiv2019

An inexact strategy for the projected gradient algorithm in vector optimization problems on variable ordered spaces

Variable order structures model situations in which the comparison between two points depends on a point-to-cone map. In this paper, an inexact projected gradient method for solving smooth constrained vector optimization problems on variable ordered spaces is presented. It is shown that every accumulation point of the generated sequence satisfies the first order necessary optimality condition. The convergence of all accumulation points to a weakly efficient point is established under suitable convexity assumptions for the objective function. The convergence results are also derived in the particular case in which the problem is unconstrained and if exact directions are taken as descent directions. Furthermore, we investigate the application of the proposed method to optimization models where the domain of the variable order map and the objective function are the same. In this case, similar concepts and convergence results are presented. Finally, some computational experiments designed to illustrate the behavior of the proposed inexact methods versus the exact ones (in terms of CPU time) are performed.

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arXiv2017

Circumcentering the Douglas--Rachford method

We introduce and study a geometric modification of the Douglas-Rach\-ford method called the Circumcentered-Douglas-Rachford method. This method iterates by taking the intersection of bisectors of reflection steps for solving certain classes of feasibility problems. The convergence analysis is established for best approximation problems involving two (affine) subspaces and both our theoretical and numerical results compare favorably to the original Douglas-Rachford method. Under suitable conditions, it is shown that the linear rate of convergence of the Circumcentered-Douglas-Rachford method is at least the cosine of the Friedrichs angle between the (affine) subspaces, which is known to be the sharp rate for the Douglas-Rachford method. We also present a preliminary discussion on the Circumcentered-Douglas-Rachford method applied to the many set case and to examples featuring non-affine convex sets.

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arXiv2016

A semi-smooth Newton method for projection equations and linear complementarity problems with respect to the second order cone

In this paper a special semi-smooth equation associated to the second order cone is studied. It is shown that, under mild assumptions, the semi-smooth Newton method applied to this equation is well-defined and the generated sequence is globally and Q-linearly convergent to a solution. As an application, the obtained results are used to study the linear second order cone complementarity problem, with special emphasis on the particular case of positive definite matrices. Moreover, some computational experiments designed to investigate the practical viability of the method are presented.

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arXiv2016

On proximal subgradient splitting method for minimizing the sum of two nonsmooth convex functions

In this paper we present a variant of the proximal forward-backward splitting iteration for solving nonsmooth optimization problems in Hilbert spaces, when the objective function is the sum of two nondifferentiable convex functions. The proposed iteration, which will be called Proximal Subgradient Splitting Method, extends the classical subgradient iteration for important classes of problems, exploiting the additive structure of the objective function. The weak convergence of the generated sequence was established using different stepsizes and under suitable assumptions. Moreover, we analyze the complexity of the iterates.

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arXiv2015

On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in real Hilbert Spaces

This work focuses on convergence analysis of the projected gradient method for solving constrained convex minimization problem in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo linesearch converges weakly to a solution of the considered convex optimization problem. Weak convergence is established by assuming convexity and Gateaux differentiability of the objective function, whose Gateaux derivative is supposed to be uniformly continuous on bounded sets. Furthermore, we propose some modifications in the classical projected gradient method in order to obtain strong convergence. The new variant has the following desirable properties: the sequence of generated points is entirely contained in a ball with diameter equal to the distance between the initial point and the solution set, and the whole sequence converges strongly to the solution of the problem that lies closest to the initial iterate. Convergence analysis of both methods is presented without Lipschitz continuity assumption.

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