Nonmonotone trust region methods with curvilinear path in unconstrained optimization

Xiao, Yi; Zhou, Fen

doi:10.1007/bf02238640

Cited by 24 publications

(11 citation statements)

References 12 publications

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“…We may also think of investigating even more efficient algorithms combining the trust-region framework developed here with other globalization techniques, like linesearches [17,31,38], non-monotone techniques [39,41,43] or filter methods [20]. While this might add yet another level of technicity to the convergence proofs, we expect such extensions to be possible and the resulting algorithms to be of practical interest.…”

Section: Comments and Perspectivesmentioning

confidence: 99%

Recursive Trust-Region Methods for Multiscale Nonlinear Optimization

Gratton¹,

Sartenaer²,

Toint³

2008

SIAM J. Optim.

159

173

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Abstract. A class of trust-region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a mean of speeding up the computation of the step. This use is recursive, leading to true multilevel/multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partial-differential equations. A simple algorithm of the class is then described and its numerical performance is shown to be numerically promising. This observation then motivates a proof of global convergence to first-order stationary points on the fine grid that is valid for all algorithms in the class.Keywords: nonlinear optimization, multiscale problems, simplified models, recursive algorithms, convergence theory.AMS subject classifications. 90C30, 65K05, 90C26, 90C061. Introduction. Large-scale finite-dimensional optimization problems often arise from the discretization of infinite-dimensional problems, a primary example being optimal-control problems defined in terms of either ordinary or partial differential equations. While the direct solution of such problems for a discretization level is often possible using existing packages for large-scale numerical optimization, this technique typically does make very little use of the fact that the underlying infinite-dimensional problem may be described at several discretization levels; the approach thus rapidly becomes cumbersome. Motivated by this observation, we explore here a class of algorithms which makes explicit use of this fact in the hope of improving efficiency and, possibly, enhancing reliability.Using the different levels of discretization for an infinite-dimensional problem is not a new idea. A simple first approach is to use coarser grids in order to compute approximate solutions which can then be used as starting points for the optimization problem on a finer grid (see [5,6,7,22], for instance). However, potentially more efficient techniques are inspired from the multigrid paradigm in the solution of partial differential equations and associated systems of linear algebraic equations (see, for example, [10,11,12,23,40,42], for descriptions and references). The work presented here was in particular motivated by the paper by Gelman and Mandel [16], the "generalized truncated Newton algorithm" presented in Fisher [15], a talk by Moré [27] and the contributions by Nash and co-authors [25,26,29]. These latter three papers present the description of MG/OPT, a linesearch-based recursive algorithm, an outline of its convergence properties and impressive numerical results. The generalized truncated Newton algorithm and MG/OPT are very similar and, like many linesearch methods, naturally suited to convex problems, although their generalization to the nonconvex case is possible. Further motivation is also provided by the computational success of the low/high-fidelity mo...

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Section: Comments and Perspectivesmentioning

confidence: 99%

Recursive Trust-Region Methods for Multiscale Nonlinear Optimization

Gratton¹,

Sartenaer²,

Toint³

2008

SIAM J. Optim.

159

173

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“…The nonmonotone strategy defined here is different from the one in [23] and [4]. In this paper, # is fixed, but in those two papers, #k was used insteading of #.…”

Section: Nonmonotone Stabilization Trust Region Algorithm (Nstr)mentioning

confidence: 98%

“…In this paper, we consider a general scheme which combines the nonmonotone trust region idea of [4] and [23] with the watchdog technique of [3]. In Section 2, a nonmonotone stabilization trust region (NSTR) algorithm is described.…”

Section: Introductionmentioning

confidence: 99%

“…Grippo, Lampariello and Lucidi [9] established the first nonmonotonic method for unconstrained optimization and developed the method relating to the line search strategy in [10], [11], [12] and [13]. The nonmonotonic methods relating to the trust region strategy were investigated by Deng,Xiao and Zhou in [4] and [23]. Some nonmonotone methods for nonlinear systems of equations were presented by Ferris and Lucidi in [6] and [7], and Xiao and Chu in [21] and [22].…”

Section: Introductionmentioning

confidence: 99%

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A class of nonmonotone stabilization trust region methods

Zhou

Xiao

1994

Computing

Self Cite

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--ZusammenfassungA Class of Nonmonotone Stabilization Trust Region Methods. A class of trust region methods in unconstrained optimization is presented, by adopting a nonmonotone stabilization strategy. Under some regularity conditions, the convergence properties of these methods are discussed. Extensive numerical results which are reported show that these methods are very efficient. AMS ( MOS) Subject Classifications: 90C30, 65K05Key words: Unconstrained optimization, trust region method, nonmonotone stabilization method. Eine Klasse yon Trust-Region-Verfahren mit niehtmonotoner Stabilisierung. Fiir das nichtrestringierteOptimierungsproblem wird eine neue Klasse yon Trust-Region-Verfahren vorgestellt, die mit einer nichtmonotonen Stabilisierungsstrategie arbeiten. Die Konvergenzeigenschaften dieser Veffahren wetden unter gewissen Regularit~itsannahmen untersucht. Umfangreiche numerische Beispiele zeigen die hohe Effizienz dieser Verfahren.

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“…Computational results 14] have shown this technique to be even better than the standard nonmonotone line search. This technique has also proven useful in other applications and extensions 6,7,8,25,26,27]. Of course, sometimes it may lead to regions where the function is poorly behaved.…”

Section: Nonmonotone Stabilization Algorithmmentioning

confidence: 99%

Nonmonotone curvilinear line search methods for unconstrained optimization

Ferris

Lucidi

Roma

1996

Comput Optim Applic

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We present a new algorithmic framework for solving unconstrained minimization problems that incorporates a curvilinear linesearch. The search direction used in our framework is a combination of an approximate Newton direction and a direction of negative curvature. Global convergence to a stationary point where the Hessian matrix is positive semide nite is exhibited for this class of algorithms by means of a nonmonotone stabilization strategy. An implementation using the Bunch-Parlett decomposition is shown to outperform several other techniques on a large class of test problems.

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Nonmonotone trust region methods with curvilinear path in unconstrained optimization

Cited by 24 publications

References 12 publications

Recursive Trust-Region Methods for Multiscale Nonlinear Optimization

Recursive Trust-Region Methods for Multiscale Nonlinear Optimization

A class of nonmonotone stabilization trust region methods

Nonmonotone curvilinear line search methods for unconstrained optimization

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