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.

A nonmonotone truncated Newton-Krylov method exploiting negative curvature directions, for large scale unconstrained optimization / Fasano, Giovanni; Lucidi, Stefano. - In: OPTIMIZATION LETTERS. - ISSN 1862-4472. - 3:4(2009), pp. 521-535. [10.1007/s11590-009-0132-y]

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

FASANO, Giovanni;LUCIDI, Stefano
2009

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.
2009
truncated newton methods; nonmonotone stabilization technique; second order necessary conditions; conjugate directions; nonlinear programming; truncated-newton methods; unconstrained optimization; negative curvatures
01 Pubblicazione su rivista::01a Articolo in rivista
A nonmonotone truncated Newton-Krylov method exploiting negative curvature directions, for large scale unconstrained optimization / Fasano, Giovanni; Lucidi, Stefano. - In: OPTIMIZATION LETTERS. - ISSN 1862-4472. - 3:4(2009), pp. 521-535. [10.1007/s11590-009-0132-y]
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