How Grossone Can Be Helpful to Iteratively Compute Negative Curvature Directions

  • Renato De Leone
  • Giovanni FasanoEmail author
  • Massimo Roma
  • Yaroslav D. Sergeyev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11353)


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.


Negative curvature directions Second order necessary optimality conditions Grossone Conjugate gradient method 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Scuola di Scienze e Tecnologie, Università di CamerinoCamerinoItaly
  2. 2.Dipartimento di ManagementUniversità Ca’ Foscari VeneziaVeniceItaly
  3. 3.Dipartimento di Ingegneria InformaticaAutomatica e Gestionale ‘A. Ruberti’, SAPIENZA, Università di RomaRomeItaly
  4. 4.Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e SistemisticaUniversità della CalabriaRendeItaly

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