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A nonmonotone truncated Newton–Krylov method exploiting negative curvature directions, for large scale unconstrained optimization

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Abstract

We propose a new truncated Newton method for large scale unconstrained optimization, where a Conjugate Gradient (CG)-based technique is adopted to solve Newton’s equation. In the current iteration, the Krylov method computes a pair of search directions: the first approximates the Newton step of the quadratic convex model, while the second is a suitable negative curvature direction. A test based on the quadratic model of the objective function is used to select the most promising between the two search directions. Both the latter selection rule and the CG stopping criterion for approximately solving Newton’s equation, strongly rely on conjugacy conditions. An appropriate linesearch technique is adopted for each search direction: a nonmonotone stabilization is used with the approximate Newton step, while an Armijo type linesearch is used for the negative curvature direction. The proposed algorithm is both globally and superlinearly convergent to stationary points satisfying second order necessary conditions. We carry out a significant numerical experience in order to test our proposal.

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Authors and Affiliations

  1. Dipartimento di Matematica Applicata, Università Ca’Foscari di Venezia, Dorsoduro 3825/e, 30123, Venice, Italy

    Giovanni Fasano

  2. Dipartimento di Informatica e Sistemistica A. Ruberti, Sapienza Università di Roma, via Ariosto 25, 00185, Rome, Italy

    Stefano Lucidi

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  1. Giovanni Fasano
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  2. Stefano Lucidi
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Correspondence to Giovanni Fasano.

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Fasano, G., Lucidi, S. A nonmonotone truncated Newton–Krylov method exploiting negative curvature directions, for large scale unconstrained optimization. Optim Lett 3, 521–535 (2009). https://doi.org/10.1007/s11590-009-0132-y

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  • DOI: https://doi.org/10.1007/s11590-009-0132-y

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