On the use of third-order models with fourth-order regularization for unconstrained optimization

Birgin, Ernesto G.; Gardenghi, John Lenon Cardoso; Martínez, José Mário; Santos, Sandra A.

doi:10.1007/s11590-019-01395-z

Cited by 15 publications

(10 citation statements)

References 27 publications

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“…2 and 3, are based on the solution of the auxiliary optimization problem (2.6). In the existing literature on the tensor methods [6,7,20,32], it was solved by the standard local technique of Nonconvex Optimization. However, now we know that by Theorem 1, this problem is convex.…”

Section: Third-order Methods: Implementation Detailsmentioning

confidence: 99%

Implementable tensor methods in unconstrained convex optimization

2019

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In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition

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Section: Third-order Methods: Implementation Detailsmentioning

confidence: 99%

Implementable tensor methods in unconstrained convex optimization

2019

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“…Consequently, while high-degree exact approaches (see [4,3,7,2] for instance) deal with a p-th degree Taylor-series approximation…”

Section: Taylor Decrements and Enforcing Accuracymentioning

confidence: 99%

“…and Lemma 2.1 (i) then ensures that the call to VERIFY in Step 1.2 returns accuracy j as sufficient, causing Algorithm 2.2 to terminate in Step 1 3…”

mentioning

confidence: 99%

Strong Evaluation Complexity of An Inexact Trust-Region Algorithm for Arbitrary-Order Unconstrained Nonconvex Optimization

Cartis,

Gould,

Toint

2020

Preprint

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A trust-region algorithm using inexact function and derivatives values is introduced for solving unconstrained smooth optimization problems. This algorithm uses high-order Taylor models and allows the search of strong approximate minimizers of arbitrary order. The evaluation complexity of finding a q-th approximate minimizer using this algorithm is then shown, under standard conditions, to be O min j∈{1,...,q} ǫ −(q+1) jwhere the ǫ j are the order-dependent requested accuracy thresholds. Remarkably, this order is identical to that of classical trust-region methods using exact information.

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“…3 ) inner iterations of M 3 if σ (k) is unbounded. In this case, a possible choice for M 3 satisfying A2' is given by [8].…”

Section: Discussionmentioning

confidence: 99%

On the Complexity of an Augmented Lagrangian Method for Nonconvex Optimization

Grapiglia,

Yuan

2019

Preprint

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In this paper we study the worst-case complexity of an inexact Augmented Lagrangian method for nonconvex constrained problems. Assuming that the penalty parameters are bounded, we prove a complexity bound of O(| log(ǫ)|) iterations for the referred algorithm generate an ǫ-approximate KKT point, for ǫ ∈ (0, 1). When the penalty parameters are unbounded, we prove an iteration complexity bound of O ǫ −2/(α−1) , where α > 1 controls the rate of increase of the penalty parameters. For linearly constrained problems, these bounds yield to evaluation complexity, respectively, when suitable p-order methods (p ≥ 2) are used to approximately solve the unconstrained subproblems at each iteration of our Augmented Lagrangian scheme.

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On the use of third-order models with fourth-order regularization for unconstrained optimization

Cited by 15 publications

References 27 publications

Implementable tensor methods in unconstrained convex optimization

Implementable tensor methods in unconstrained convex optimization

Strong Evaluation Complexity of An Inexact Trust-Region Algorithm for Arbitrary-Order Unconstrained Nonconvex Optimization

On the Complexity of an Augmented Lagrangian Method for Nonconvex Optimization

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