Abstract
We consider an iterative computation of negative curvature directions, in large scale optimization frameworks. We show that to the latter purpose, borrowing the ideas in [1, 3] and [4], we can fruitfully pair the Conjugate Gradient (CG) method with a recently introduced numerical approach involving the use of grossone [5]. In particular, though in principle the CG method is well-posed only on positive definite linear systems, the use of grossone can enhance the performance of the CG, allowing the computation of negative curvature directions, too. The overall method in our proposal significantly generalizes the theory proposed for [1] and [3], and straightforwardly allows the use of a CG-based method on indefinite Newton’s equations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Fasano, G.: Conjugate Gradient (CG)-type method for the solution of Newton’s equation within optimization frameworks. Optim. Methods Softw. 19(3–4), 267–290 (2004)
Fasano, G.: Planar-conjugate gradient algorithm for large scale unconstrained optimization, part 1: theory. J. Optim. Theory Appl. 125(3), 523–541 (2005)
Fasano, G., Roma, M.: Iterative computation of negative curvature directions in large scale optimization. Comput. Optim. Appl. 38(1), 81–104 (2007)
De Leone, R., Fasano, G., Sergeyev, Y.D.: Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming. Comput. Optim. Appl. 71(1), 73–93 (2018)
Sergeyev, Y.D.: Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems. EMS Surv. Math. Sci. 4, 219–320 (2017)
De Leone, R.: Nonlinear programming and grossone: quadratic programming and the role of constraint qualifications. Appl. Math. Comput. 318, 290–297 (2018)
Gaudioso, M., Giallombardo, G., Mukhametzhanov, M.: Numerical infinitesimals in a variable metric method for convex nonsmooth optimization. Appl. Math. Comput. 318, 312–320 (2018)
Sergeyev, Y.D., Kvasov, D.E., Mukhametzhanov, M.: On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales. Commun. Nonlinear Sci. Numer. Simul. 59, 319–330 (2018)
Moré, J., Sorensen, D.: On the use of directions of negative curvature in a modified Newton method. Math. Program. 16, 1–20 (1979)
De Leone, R., Fasano, G., Roma, M., Sergeyev, Y.D.: Iterative
-based computation of negative curvature directions in large scale optimization (submitted)
-based computation of negative curvature directions in large scale optimization (submitted)Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
De Leone, R., Fasano, G., Roma, M., Sergeyev, Y.D. (2019). How Grossone Can Be Helpful to Iteratively Compute Negative Curvature Directions. In: Battiti, R., Brunato, M., Kotsireas, I., Pardalos, P. (eds) Learning and Intelligent Optimization. LION 12 2018. Lecture Notes in Computer Science(), vol 11353. Springer, Cham. https://doi.org/10.1007/978-3-030-05348-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-030-05348-2_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05347-5
Online ISBN: 978-3-030-05348-2
eBook Packages: Computer ScienceComputer Science (R0)