Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming
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This paper deals with an analysis of the Conjugate Gradient (CG) method (Hestenes and Stiefel in J Res Nat Bur Stand 49:409–436, 1952), in the presence of degenerates on indefinite linear systems. Several approaches have been proposed in the literature to issue the latter drawback in optimization frameworks, including reformulating the original linear system or recurring to approximately solving it. All the proposed alternatives seem to rely on algebraic considerations, and basically pursue the idea of improving numerical efficiency. In this regard, here we sketch two separate analyses for the possible CG degeneracy. First, we start detailing a more standard algebraic viewpoint of the problem, suggested by planar methods. Then, another algebraic perspective is detailed, relying on a novel recently proposed theory, which includes an additional number, namely grossone. The use of grossone allows to work numerically with infinities and infinitesimals. The results obtained using the two proposed approaches perfectly match, showing that grossone may represent a fruitful and promising tool to be exploited within Nonlinear Programming.
KeywordsConjugate Gradient (CG) method Planar-CG methods Infinities and Infinitesimals Grossone
G. Fasano thanks the National Research Council-Marine Technology Research Institute (CNR-INSEAN), Italy, for the support received. The work of G. Fasano is partially supported by the Italian Flagship Project RITMARE, coordinated by the Italian National Research Council (CNR) and funded by the Italian Ministry of Education, within the National Research Program 2012–2016. The research of Ya.D. Sergeyev was supported by the Russian Science Foundation, Project No. 15-11-30022 “Global optimization, supercomputing computations, and applications”.
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