Conditional subgradient optimization — Theory and applications

Larsson, Torbjörn; Patriksson, Michael; Strömberg, Ann-Brith

doi:10.1016/0377-2217(94)00200-2

Cited by 69 publications

(51 citation statements)

References 26 publications

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“…A (ε-)subgradient of f X is said to be a conditional (ε)-subgradient of f w.r.t. X [22,24] and can be used instead of g k to compute the search direction. It is well known that ∂ ε f X (x) ⊇ ∂ ε f (x)+∂I X (x) = ∂ ε f (x)+N k [18], where N k = N X (x k ) is the normal cone of X at x k ; thus, for iterates x k on the frontier of X (where N k = {0}), one may have, for a given g k produced by the "black box," multiple choices of vectors in the normal cone to produce a conditional subgradient g k to be used for computing the direction.…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Deflected Conditional Approximate Subgradient Methods

d'Antonio¹,

Frangioni²

2009

SIAM J. Optim.

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Abstract. Subgradient methods for nondifferentiable optimization benefit from deflection, i.e., defining the search direction as a combination of the previous direction and the current subgradient. In the constrained case they also benefit from projection of the search direction onto the feasible set prior to computing the steplength, that is, from the use of conditional subgradient techniques. However, combining the two techniques is not straightforward, especially if an inexact oracle is available which can only compute approximate function values and subgradients. We present a convergence analysis of several different variants, both conceptual and implementable, of approximate conditional deflected subgradient methods. Our analysis extends the available results in the literature by using the main stepsize rules presented so far, while allowing deflection in a more flexible way. Furthermore, to allow for (diminishing/square summable) rules where the stepsize is tightly controlled a priori, we propose a new class of deflection-restricted approaches where it is the deflection parameter, rather than the stepsize, which is dynamically adjusted using the "target value" of the optimization sequence. For both Polyak-type and diminishing/square summable stepsizes, we propose a "correction" of the standard formula which shows that, in the inexact case, knowledge about the error computed by the oracle (which is available in several practical applications) can be exploited in order to strengthen the convergence properties of the method. The analysis allows for several variants of the algorithm; at least one of them is likely to show numerical performances similar to these of "heavy ball" subgradient methods, popular within backpropagation approaches to train neural networks, while possessing stronger convergence properties.

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Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Deflected Conditional Approximate Subgradient Methods

d'Antonio¹,

Frangioni²

2009

SIAM J. Optim.

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“…For a discussion of subgradient optimization strategies see Sarin and Karwan (1987), Sherali and Myers (1988), and Baker and Sheasby (1999). The use of conditional subgradient optimization is analyzed in Larsson et al (1996). The choice of the step size in subgradient optimization algorithms is addressed by Myers andby Poljak (1967, 1969), Held et al (1974) and Bazaraa and Sherali (1981).…”

Section: A Brief Review Of Lagrangean Relaxationmentioning

confidence: 99%

Hub Location Problems: The Location of Interacting Facilities

Kara

Taner

2011

International Series in Operations Research &Amp; Management Science

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), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. PrefaceThis book is the final result of a number of incidents that occurred to us while discussing location issues with colleagues at conferences. Frequently we attended presentations, in which the authors quoted the well-known references that helped to make the discipline what it is today. Upon further inquiry, though, it turned out that some of these colleagues had never actually read the original papers. We then discussed among ourselves which contributions could be credited with shaping the field. And, lo and behold, we found that we, too, had neglected to read some of the papers that form the foundation of our science. Whether it was laziness or other things that got in the way, it had become clear that something had to be done. Our first thought was to collect the original contributions (once we could agree on what they were) and reprint them. When discussing this possibility with a publisher, we immediately ran into a roadblock in the form of copyright. While this appeared to have stopped our enthusiastic effort dead in its tracks, we kept on collecting and reading what we considered original contributions.This went on until we met Camille Price, who suggested that, rather than reprinting the original contributions, we should invite some of the leaders in our field and ask them to describe the original contribution, explain and interpret it, and comment on the impact that it had to the field. This was, of course, an excellent idea, and the response by our colleagues to our pertinent requests was equally enthusiastic. What you hold in your hands is the result of this effort.In other words, the purpose of this book is to provide easy access to the main contributions to location theory. The book is organized as follows. The introductory chapter provides an overview of some of the many facets of location analysis. This is followed by contributions in the main three fields of inquiry: minisum, minimax, and covering problems. The next chapters are part of an ever-growing list of nonstandard location models: models including competitive components, those that locate undesirable facilities, those with probabilistic features, and those that allow interactions between facilities. The following chapters discuss solution techniques: after a discussion of exact and heuristic techniques, we devote an entire chapter to Weiszfeld's method, and another to Lagrangean techniques. The last chapters of this book deal with the spheres of influence that the facilities generate and that attract viii customers to them, an...

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“…There are many other subgradient methods in the literature [4,5,6,23,24,26]. They increase the local computational times computing descent directions [6], or combining subgradients of previous iterations [4,5], or realizing projections onto general convex sets [23,24,26].…”

Section: Introductionmentioning

confidence: 99%

“…They increase the local computational times computing descent directions [6], or combining subgradients of previous iterations [4,5], or realizing projections onto general convex sets [23,24,26]. Experimental results with some of these methods show an improvement in performance compared to the subgradient method [23,26], but the subgradient method remains the widely used approach in the Lagrangean relaxation context.…”

Section: Introductionmentioning

confidence: 99%

Using logical surrogate information in Lagrangean relaxation: An application to symmetric traveling salesman problems

Lorena

Narciso

2002

European Journal of Operational Research

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. AbstractThe Traveling Salesman Problem (TSP) is a classical Combinatorial Optimization problem, which has been intensively studied. The Lagrangean relaxation was first applied to the TSP in 1970. The Lagrangean relaxation limit approximates what is known today as HK (Held and Karp) bound, a very good bound (less than 1% from optimal) for a large class of symmetric instances. It became a reference bound for new heuristics, mainly for the very large scale instances, where the use of exact methods is prohibitive. A known problem for the Lagrangean relaxation application is the definition of a convenient step size control in subgradient like methods. Even preserving theoretical convergence properties, a wrong defined control can affect the performance and increase computational times. We show in this work how to accelerate a classical subgradient method while conserving good approximations to the HK bounds. The surrogate and Lagrangean relaxation are combined using the local information of the relaxed constraints. It results in a one-dimensional search that corrects the possibly wrong step size and is independent of the used step size control. Comparing with the ordinary subgradient method, and beginning with the same initial multiplier, the computational times are almost twice as fast for medium instances and greatly improved for some large scale TSPLIB instances.

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Conditional subgradient optimization — Theory and applications

Cited by 69 publications

References 26 publications

Convergence Analysis of Deflected Conditional Approximate Subgradient Methods

Convergence Analysis of Deflected Conditional Approximate Subgradient Methods

Hub Location Problems: The Location of Interacting Facilities

Using logical surrogate information in Lagrangean relaxation: An application to symmetric traveling salesman problems

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