A tensor trust-region model for nonlinear system

Wang, Songhua; Liu, Shulun

doi:10.1186/s13660-018-1935-0

Cited by 2 publications

(3 citation statements)

References 40 publications

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“…Let us first consider the case in which the Taylor model is employed. It follows from (31), (24) and ( 29) that for all…”

Section: Local Convergencementioning

confidence: 99%

See 1 more Smart Citation

On high-order multilevel optimization strategies

Calandra¹,

Gratton²,

Riccietti³

et al. 2019

Preprint

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We propose a new family of multilevel methods for unconstrained minimization. The resulting strategies are multilevel extensions of high-order optimization methods based on q-order Taylor models (with q ≥ 1) that have been recently proposed in the literature. The use of high-order models, while decreasing the worst-case complexity bound, makes these methods computationally more expensive. Hence, to counteract this effect, we propose a multilevel strategy that exploits a hierarchy of problems of decreasing dimension, still approximating the original one, to reduce the global cost of the step computation. A theoretical analysis of the family of methods is proposed. Specifically, local and global convergence results are proved and a complexity bound to reach first order stationary points is also derived. A multilevel version of the well known adaptive method based on cubic regularization (ARC, corresponding to q = 2 in our setting) has been implemented. Numerical experiments clearly highlight the relevance of the new multilevel approach leading to considerable computational savings in terms of floating point operations compared to the classical one-level strategy.

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“…Let us first consider the case in which the Taylor model is employed. It follows from (31), (24) and ( 29) that for all…”

Section: Local Convergencementioning

confidence: 99%

“…In recent publications also methods of higher order start to gain interest, see for example [2,24]. In [2] in particular, it has been observed that the good complexity bound of ARC can be made even lower, if one is willing to use higher-order derivatives.…”

Section: Introductionmentioning

confidence: 99%

On high-order multilevel optimization strategies

Calandra¹,

Gratton²,

Riccietti³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…In recent publications also methods of higher order start to gain interest, see for example [6,41]. In [6] in particular, it has been observed that the good complexity bound of ARC can be made even lower, if one is willing to use higher-order derivatives.…”

mentioning

confidence: 99%

On High-Order Multilevel Optimization Strategies

Calandra¹,

Gratton²,

Riccietti³

et al. 2021

SIAM J. Optim.

View full text Add to dashboard Cite

We propose a new family of multilevel methods for unconstrained minimization. The resulting strategies are multilevel extensions of high-order optimization methods based on q-th order Taylor models (with q ≥ 1) that have been recently proposed in the literature. The use of high-order models, while decreasing the worst-case complexity bound, makes these methods computationally more expensive. Hence, to counteract this effect, we propose a multilevel strategy that exploits a hierarchy of problems of decreasing dimension, still approximating the original one, to reduce the global cost of the step computation. A theoretical analysis of the family of methods is proposed. Specifically, local and global convergence results are proved and a worst-case complexity bound to reach first-order stationary points is also derived. A multilevel version of the well-known adaptive regularization by cubics (corresponding to q = 2 in our setting) has been implemented, as well as a multilevel third-order method (q = 3). Numerical experiments clearly highlight the relevance of the new multilevel approaches leading to considerable computational savings compared to their one-level counterparts.

show abstract

A tensor trust-region model for nonlinear system

Cited by 2 publications

References 40 publications

On high-order multilevel optimization strategies

On high-order multilevel optimization strategies

On High-Order Multilevel Optimization Strategies

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