Nonconvex optimization using negative curvature within a modified linesearch

Olivares, Alberto; Moguerza, Javier M.; Prieto, Francisco J.

doi:10.1016/j.ejor.2006.09.097

Cited by 16 publications

(13 citation statements)

References 23 publications

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“…The rational behind using a different linesearch technique for each direction, is the possibility of capturing possible differences between the two directions. In [20] there is an attempt to match both the approaches above, in order to yield an efficient algorithm for small scale problems, adopting a monotone stabilization technique.…”

Section: Introductionmentioning

confidence: 98%

“…In particular, we observe that the state-of-the-art Newton-type methods are based on the idea of exploiting the local information on the function f (x), obtained by investigating the second order derivatives. In the context of large scale problems, the latter task is pursued by computing at the outer iteration k the pair (d k , s k ) of promising search directions [5,7,8,14,17,20], by means of efficient iterative techniques. Roughly speaking, d k summarizes the local convexity of f (x) at the current iterate, while s k takes into account the local nonconvexity of the objective function.…”

Section: Introductionmentioning

confidence: 99%

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

Fasano

Lucidi

2009

Optim Lett

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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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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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

Fasano

Lucidi

2009

Optim Lett

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“…In [9] the alternative use of a negative curvature direction and a Newton-type direction was proposed, within an appropriate linesearch procedure. In the work of Olivares et al [17] the authors established criteria such that at each iteration, either a linesearch procedure using one direction (a Newton-type or a negative curvature direction) or a curvilinear search combining both directions, is performed. Curvilinear paths and negative curvature directions are also used in constrained optimization [16,18,19].…”

Section: ð1:2þmentioning

confidence: 99%

A curvilinear method based on minimal-memory BFGS updates

Apostolopoulou

Sotiropoulos

Botsaris³

2010

Applied Mathematics and Computation

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a b s t r a c tWe present a new matrix-free method for the computation of negative curvature directions based on the eigenstructure of minimal-memory BFGS matrices. We determine via simple formulas the eigenvalues of these matrices and we compute the desirable eigenvectors by explicit forms. Consequently, a negative curvature direction is computed in such a way that avoids the storage and the factorization of any matrix. We propose a modification of the L-BFGS method in which no information is kept from old iterations, so that memory requirements are minimal. The proposed algorithm incorporates a curvilinear path and a linesearch procedure, which combines two search directions; a memoryless quasi-Newton direction and a direction of negative curvature. Results of numerical experiments for large scale problems are also presented.

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“…Update the parameters; k = k + 1 Until Convergence There are some works describing the advantages of using a "good" direction of negative curvature (see for example Prieto 2003a, 2003b;Gould et al 2000or Olivares et al 2008. In particular, in Olivares et al (2008) an adapted procedure for choosing the best combination of directions is proposed. In this work it is shown that in some iterations it is better to combine a Newton direction and a direction of negative curvature than to discard one of them.…”

Section: Nonlinear Optimization Algorithm Using Directions Of Negativmentioning

confidence: 99%

Improving directions of negative curvature in an efficient manner

Olivares

Moguerza

2008

Ann Oper Res

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In order to converge to second-order KKT points, second derivative information has to be taken into account. Therefore, methods for minimization satisfying convergence to second-order KKT points must, at least implicitly, compute a direction of negative curvature of an indefinite matrix. An important issue is to determine the quality of the negative curvature direction. This problem is closely related to the symmetric eigenvalue problem. More specifically we want to develop algorithms that improve directions of negative curvature with relatively little effort, extending the proposals by Boman and Murray. This paper presents some technical improvements with respect to their work. In particular, we study how to compute "good" directions of negative curvature. In this regard, we propose a new method and we present numerical experiments that illustrate its practical efficiency compared to other proposals.

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Nonconvex optimization using negative curvature within a modified linesearch

Cited by 16 publications

References 23 publications

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

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

A curvilinear method based on minimal-memory BFGS updates

Improving directions of negative curvature in an efficient manner

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