Bundle Methods for Inexact Data

Oliveira, Welington de; Solodov, M. V.

doi:10.1007/978-3-030-34910-3_12

Cited by 7 publications

(2 citation statements)

References 70 publications

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“…In practice, bundle methods have proven to be efficient methods for solving many nonsmooth problems (see [6,35,36] for further discussion). Extensions that apply to nonconvex problems have been considered in in [2,5,12,17,[27][28][29][30] and as well as an extension to problems where only an inexact first-order oracle is available in [7,13,26].…”

Section: Related Workmentioning

confidence: 99%

Optimal Convergence Rates for the Proximal Bundle Method

Díaz¹,

Grimmer²

2021

Preprint

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We study convergence rates of the classic proximal bundle method for a variety of nonsmooth convex optimization problems. We show that, without any modification, this algorithm adapts to converge faster in the presence of smoothness or a Hölder growth condition. Our analysis reveals that with a constant stepsize, the bundle method is adaptive, yet it exhibits suboptimal convergence rates. We overcome this shortcoming by proposing nonconstant stepsize schemes with optimal rates. These schemes use function information such as growth constants, which might be prohibitive in practice. We complete the paper with a new parallelizable variant of the bundle method that attains near-optimal rates without prior knowledge of function parameters. These results improve on the limited existing convergence rates and provide a unified analysis approach across problem settings and algorithmic details. Numerical experiments support our findings and illustrate the effectiveness of the parallel bundle method.

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Section: Related Workmentioning

confidence: 99%

Optimal Convergence Rates for the Proximal Bundle Method

Díaz¹,

Grimmer²

2021

Preprint

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“…Inexact evaluations are usually referred to in stochastic programming and Lagrangian relaxation [30,31]. It is at the very least impractical and is usually not even easy to solve those subproblems exactly.…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Proximal Bundle Method with Inexact Oracles for a Class of Nonconvex and Nonsmooth Composite Optimization

Wang

Pang

et al. 2021

Mathematics

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In this paper, an adaptive proximal bundle method is proposed for a class of nonconvex and nonsmooth composite problems with inexact information. The composite problems are the sum of a finite convex function with inexact information and a nonconvex function. For the nonconvex function, we design the convexification technique and ensure the linearization errors of its augment function to be nonnegative. Then, the sum of the convex function and the augment function is regarded as an approximate function to the primal problem. For the approximate function, we adopt a disaggregate strategy and regard the sum of cutting plane models of the convex function and the augment function as a cutting plane model for the approximate function. Then, we give the adaptive nonconvex proximal bundle method. Meanwhile, for the convex function with inexact information, we utilize the noise management strategy and update the proximal parameter to reduce the influence of inexact information. The method can obtain an approximate solution. Two polynomial functions and six DC problems are referred to in the numerical experiment. The preliminary numerical results show that our algorithm is effective and reliable.

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