On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization

Solodov, M. V.

doi:10.1023/b:jota.0000005046.70410.02

Cited by 36 publications

(28 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…First note that if the oracle errors do not vanish in the limit, of course only approximate solutions to (1) can be expected in general. This is natural, and similar to [27,18,24,23].…”

Section: Handling Inexact Datamentioning

confidence: 62%

“…Various inexact bundle methods that use approximate function and subgradient evaluations have been studied in [14,27,18,24,23]. The natural setting is to assume that, given any x ∈ ℜ n , the oracle provides some approximate values f x ∈ ℜ and g x ∈ ℜ n of the objective function and its subgradient, respectively, such that…”

Section: Handling Inexact Datamentioning

confidence: 99%

“…(27) and Step 5 employing (29) with µ max 1, there exists an index k 1 1 such that the descent test (25) is not satisfied for all k k 1 .…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Suppose first that there exists an infinite subsequence of null proximal steps (µ k = 1), indexed by {k( j)}. "Ignoring" the possible null level steps inbetween, we can consider the sequence {x k( j) } as that generated by the proximal bundle method, where the model satisfies, by the rule (27), the key conditions…”

Section: Lemma 7 Suppose That For Algorithm 2 With the Additional Bumentioning

confidence: 99%

See 3 more Smart Citations

A doubly stabilized bundle method for nonsmooth convex optimization

Oliveira

Solodov

2015

Math. Program.

View full text Add to dashboard Cite

We propose a bundle method for minimizing nonsmooth convex functions that combines both the level and the proximal stabilizations. Most bundle algorithms use a cutting-plane model of the objective function to formulate a subproblem whose solution gives the next iterate. Proximal bundle methods employ the model in the objective function of the subproblem, while level methods put the model in the subproblem's constraints. The proposed algorithm defines new iterates by solving a subproblem that employs the model in both the objective function and in the constraints. One advantage when compared to the proximal approach is that the level set constraint provides a certain Lagrange multiplier, which is used to update the proximal parameter in a novel manner. We also show that in the case of inexact function and subgradient evaluations, no additional procedure needs to be performed by our variant to deal with inexactness (as opposed to the proximal bundle methods that require special modifications). Numerical experiments on almost one thousand instances of different types of problems are presented. Our experiments show that the doubly stabilized bundle method inherits useful features of the level and the proximal versions, and compares favorably to both of them.Keywords nonsmooth optimization · proximal bundle method · level bundle method · inexact oracle.Mathematics Subject Classification (2010) 90C25 · 90C30 · 65K05.

show abstract

“…First note that if the oracle errors do not vanish in the limit, of course only approximate solutions to (1) can be expected in general. This is natural, and similar to [27,18,24,23].…”

Section: Handling Inexact Datamentioning

confidence: 62%

Section: Handling Inexact Datamentioning

confidence: 99%

“…(27) and Step 5 employing (29) with µ max 1, there exists an index k 1 1 such that the descent test (25) is not satisfied for all k k 1 .…”

Section: ⊓ ⊔mentioning

confidence: 99%

Section: Lemma 7 Suppose That For Algorithm 2 With the Additional Bumentioning

confidence: 99%

See 2 more Smart Citations

A doubly stabilized bundle method for nonsmooth convex optimization

Oliveira

Solodov

2015

Math. Program.

View full text Add to dashboard Cite

show abstract

“…It was only in 2001 that the work [17] opened the way to inexactness. It was followed by [53] and [27].…”

Section: Introductionmentioning

confidence: 99%

BUNDLE METHODS IN THE XXIst CENTURY: A BIRD'S-EYE VIEW

Oliveira

Sagastizábal²

2014

Pesqui. Oper.

View full text Add to dashboard Cite

ABSTRACT. Bundle methods are often the algorithms of choice for nonsmooth convex optimization, especially if accuracy in the solution and reliability are a concern. We review several algorithms based on the bundle methodology that have been developed recently and that, unlike their forerunner variants, have the ability to provide exact solutions even if most of the time the available information is inaccurate. We adopt an approach that is by no means exhaustive, but covers different proximal and level bundle methods dealing with inexact oracles, for both unconstrained and constrained problems.

show abstract

Chance-Constrained Programming

2013

Encyclopedia of Operations Research and Management Science

View full text Add to dashboard Cite

In a way the basis for this thesis was laid in 2005 when together with some colleagues we were quantifying the impact of uncertainty on weekly unit-commitment. A fruitful collaboration with Pierre Thomas and Riadh Zorgati led to the birth of a software package called LEA ("Laboratoire des Expérimentations des Aléas") in the same year. From a technical perspective this software package allowed the user to execute a simple multiscenario approach around a deterministic optimization kernel. The innovating part was set up with Bernard Beauzamy. Instead of drawing scenarios as iid samples from some law (or black box software), the idea was to build artificial englobing scenarios allowing for bounding of output with only little computational effort. These experiments led, only much later, to an industrial software package. In any case this is how together with Riadh Zorgati, I got interested in dealing with uncertainty in an appropriate way. We had many discussions on Scenario generation, Robust Optimization etc These discussions were also greatly nourished by discussions with Michel Minoux. Around the same time, we had the chance to meet René Henrion and attend a seminar he gave. Chance constrained programming looked like a very powerful and intuitive way to deal with uncertainty. With good hopes and enthusiasm we set out to look where uncertainty in Energy Management plays a key role and should be dealt with appropriately. Once these structures identified we started experimenting with safe tractable approximations of chance constraints, scenario type of approaches and also Robust Optimization. Somewhere in 2009 we had to chance to work closely together with René Henrion and look to what extent Chance Constrained programming could be applied in Energy Management. This led me to be greatly intrigued by this field, and ultimately, gave me a desire to deeply understand this domain. Thanks to EDF I have had the opportunity to do this Thesis in part time. I hope to have contributed to the domain and continue to do so in the future. I do firmly believe that Chance Constrained Programming will be important in Energy Management in the near future.Thanking people after 10 years of work at EDF is almost a dangerous undertaking since we are bound to forget someone with whom we have had interesting discussions. Nonetheless I will make this attempt at my own risks.First of all I would like to thank Riadh Zorgati, who is a good friend and colleague. We share many views and in particular that (fundamental) research should have its place in an(y) industrial company. I would also like to thank Grace Doukopoulos, Olivier Feron, Nadia Oudjane, Cyrille Strugarek and Xavier Warin for the many interesting discussions (and years of joint work). Thanks to Thomas Triboulet, Laurent Mulot and their pragmatic viewpoint of what at EDF we commonly call "le métier" it is possible to obtain a partial view of the underlying structures of the optimization problems at hand. This allows for important progress in identifying the key features of Energy Ma...

show abstract

On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization

Cited by 36 publications

References 16 publications

A doubly stabilized bundle method for nonsmooth convex optimization

A doubly stabilized bundle method for nonsmooth convex optimization

BUNDLE METHODS IN THE XXIst CENTURY: A BIRD'S-EYE VIEW

Chance-Constrained Programming

Contact Info

Product

Resources

About