A descent method with linear programming subproblems for nondifferentiable convex optimization

Kim, Sehun; Chang, Kun-Nyeong; Lee, Jun-Yeon

doi:10.1007/bf01592242

Cited by 16 publications

(15 citation statements)

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“…These methods are generally called Trust Region Methods. Another approach consists in penalizing the distance separating a new dual solution from the previous one [38]. More flexible approaches are proposed in [9,20] to stabilize the column generation combining in a certain way the concept of box and the penalty method of [38].…”

Section: Related Workmentioning

confidence: 99%

“…Another approach consists in penalizing the distance separating a new dual solution from the previous one [38]. More flexible approaches are proposed in [9,20] to stabilize the column generation combining in a certain way the concept of box and the penalty method of [38]. Finally, if we have a priori information on the optimal dual problem, we can use it to restrict the dual master problem (see, e.g., [32]).…”

Section: Related Workmentioning

confidence: 99%

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Acceleration of cutting‐plane and column generation algorithms: Applications to network design

Ben‐Ameur

Neto

2006

Networks

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Most of integer, convex, and large-scale linear problems are solved using cutting plane and column generation algorithms. Therefore, to handle large-size problems and to reduce the computing times, it may be very useful to accelerate cutting plane algorithms. We show in this article that we can achieve this goal by choosing good separation points. Focus is given on problems for which we have an exact separation oracle. An in-out algorithm is proposed, and the convergence is proved under some general assumptions. Computational experiments related to three classes of problems, survivable network design, multicommodity flow problems, and random linear programs, clearly point out the savings of time allowed by the simple in-out approach proposed in this article.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Acceleration of cutting‐plane and column generation algorithms: Applications to network design

Ben‐Ameur

Neto

2006

Networks

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“…In [Au87], (1.9) is only used to prove [Au87, Theorem 2.3]. In other cases duality was completely overlooked, even when linear duality could have been used [KCL95]. A first step towards this development was done in [Fr97], where D * t = 1 t · p with p ∈ {1, 2, ∞} was studied; due to the interpretation of (1.9) in terms of ε-subgradients, those bundle variants had an interest on their own, as a bundle algorithm with a dual trust region was one of the open questions in [HL93b, Remark XV.2.5.1].…”

Section: Comparisonsmentioning

confidence: 99%

“…Dually, a penalty term must increase as the penalty parameter does (see (P * 4)), and it must be equivalent to the constraints it replaced, at least in the limit (see (P * 5)). D t need not be "norm-like" [KCL95,Be96] or a Bregman distance [CT93]; in particular, it is not necessary that…”

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confidence: 99%

“…Our conditions on D t are less restrictive than those in [Au87, KCL95,Be96], are different from those in [Ki99], and allow a unified treatment of "penalty-like" and "trust-region-like" stabilizing terms [HL93b, sections XV.2.1 and XV.2.2], which have so far been considered as distinct. Very little regularity is required for D t (d) as a function of t. Weak requirements on f , such as * -compactness, avoid stronger requirements on D t .…”

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confidence: 99%

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Generalized Bundle Methods

Frangioni¹

2002

SIAM J. Optim.

137

136

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Abstract. We study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties. This setting covers several algorithms from the literature that have been so far regarded as distinct. Under a different hypothesis on the stabilizing term and/or the function to be minimized, we prove finite termination, asymptotic convergence, and finite convergence to an optimal point, with or without limits on the number of serious steps and/or requiring the proximal parameter to go to infinity. The convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm, i.e., the management of the bundle of subgradients (β-strategy) and of the proximal parameter (t-strategy). We extensively exploit a dual view of bundle methods, which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space, where nonlinear subproblems are approximately solved at each step with an inner linearization approach. This allows us to precisely characterize the changes in the subproblems during the serious steps, since the dual problem is not tied to the local concept of ε-subdifferential. For some of the proofs, a generalization of inf-compactness, called * -compactness, is required; this concept is related to that of asymptotically well-behaved functions.

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