Other Articles by Authors

J. Arroyo
O. J. Garay
J. Mencia

Authors and WSEAS

J. Arroyo
O. J. Garay
J. Mencia

WSEAS Transactions on Mathematics

Print ISSN: 1109-2769
E-ISSN: 2224-2880

Volume 17, 2018

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

Volume 17, 2018

Quadratic Gradient Descent Methods and Transversality Conditions

AUTHORS: J. Arroyo, O. J. Garay, J. Mencia

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ABSTRACT: Different gradient descent methods have been introduced in [1] to study a quite general family of variational problems under affine and isoperimetric constraints. Thus, in [1] gradient descent sequences are created by using linear, quadratic or cubic approximations to the gradient descent trajectories, and methods are numerically implemented in a computational platform (which we call XEL-platform). In this work, performance of the above quadratic gradient descent versions are analyzed under the influence of transversality constraints. We see that, within this context, transversality conditions can be dealt as isoperimetric constraints and, then, the XEL-platform can be used to localize minimizers in the spaces of curves determined by the prescribed constraints. Efficiency of the approach is analyzed by considering two very well known classical problems, the brachystochrone and closed planar elastica. In the first case, the effect of introducing an isoperimetric constraint is also considered and the estimated errors are shown to be numerically insignificant. In the second case, minimizers are well known (circles and eight-figure curves) and we see how the XEL-platform takes very distant curves (from the energy point of view) within the same homotopy class to the minimizer included in that class. It is also capable to detect local minimizers which may appear during the gradient descent from the initial curve towards the minimum.

KEYWORDS: Gradient descent, energy minimizers, isoperimetric conditions, tranversality


[1] J. Arroyo, O.J. Garay, J.J. Menc´ıa and A. Pampano, A gradient-descent method for La- ´ grangian densities depending on multiple derivatives, Preprint, 2018.

[2] J. Arroyo, O.J. Garay and A. Pampano, Bound- ´ ary value problems for Euler-Bernoulli planar elastica. A solution construction procedure, Submitted.

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WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 17, 2018, Art. #23, pp. 178-187

Copyright © 2018 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0

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