Auxiliary problem principle and decomposition of optimization problems

Cohen, Guy

doi:10.1007/bf00934554

Cited by 334 publications

(193 citation statements)

References 7 publications

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“…Section 2.2 (strong monotony should replace monotony, the latter weaker assumption being considered in Ref. 2) and the situation when • is not symmetric but still strongly monotone as a whole. It also covers situation (d) as shown by the following lemma.…”

Section: Ucu Fmentioning

confidence: 99%

“…As we shall see later on, this case of saddle-point problems appears to be a very specific case where the magnitude of the off-diagonal terms can be, in some sense, arbitrarily large and where the (positive or negative) monotony of the block-diagonal terms is sufficient to prove convergence of algorithms such as those of Uzawa, Arrow-Hurwicz or other more general algorithms (Ref. 2).…”

Section: Introductionmentioning

confidence: 99%

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Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms

Cohen

Chaplais

1988

J Optim Theory Appl

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Abstract. The auxiliary problem principle has been proposed by the first author as a framework to describe and analyze iterative algorithms such as gradient as well as decomposition/coordination algorithms for optimization problems (Refs. 1-3) and variational inequalities (Ref. 4). The key assumption to prove the global and strong convergence of such algorithms, as well as of most of the other algorithms proposed in the literature, is the strong monotony of the operator involved in the variational inequalities. In this paper, we consider variational inequalities defined over a product of spaces and we introduce a new property of strong nested monotony, which is weaker than the ordinary overall strong monotony generally assumed. In some sense, this new concept seems to be a minimal requirement to insure convergence of the algorithms alluded to above. A convergence theorem based on this weaker assumption is given. Application of this result to the computation of Nash equilibria can be found in another paper (Ref. 5).

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Section: Ucu Fmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms

Cohen

Chaplais

1988

J Optim Theory Appl

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“…System y computes a movement direction for problem (A.5), using Newton's method, Problem (A1)-(A3) has also been solved using a Lagrangian relaxation procedure [16] and an augmented Lagrangian relaxation one [3,7]. The Lagrangian relaxation procedure uses a simple subgradient updating of multipliers, and the augmented Lagrangian decomposition procedure uses a progressively increasing penalty term and a simple gradient multiplier updating technique.…”

Section: Appendix: Examplementioning

confidence: 99%

“…Particularly remarkable is the theoretical decomposition framework based on the auxiliary problem principIe ana1yzed in [6][7][8]. An application ofLagrangian relaxation to solve a multi-area OPF is described in [1,9], while in [2,14] an augmented Lagrangian relaxation procedure is used to sol ve a distributed OPF.…”

Section: Introductionmentioning

confidence: 99%

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Nogales

Prieto

Conejo

2003

Annals of Operations Research

177

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Abstract. This paper describes a decomposition methodology applied to the multi-area optimal power fiow problem in the context of an electric energy system. The proposed procedure is simple and efficient, and presents sorne advantages with respect to other common decomposition techniques such as Lagrangian relaxation and augmented Lagrangian decomposition. The application to the multi-area optimal power fiow problem allows the computation of an optimal coordinated but decentralized solution. The proposed method is appropriate for an Independent System Operator in charge of the electric energy system technical operation. Convergence properties of the proposed decomposition algorithm are described and related to the physical coupling between the areas. Theoretical and numerical results show that the proposed decentralized methodology has a lower computational cost than other decomposition techniques, and in large large-scale cases even lower than a centralized approach.

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“…S orne authors replaced the classical metric by a nonlinear fixed metric, based on an entropie method (G. Chen and M. Teboulle [4], J. Eckstein [7], S. Kabbadj [8], ...). Others let the metric change at each itération (G. Cohen [5] and [6], M. Qian [11] and [12], J. R Bonnans, J. C. Gilbert, C. Lemaréchal and C. Sagastizabal [3], A. Renaud [13], ...).…”

Section: P Alexandre V H Nguyen P Tossingsmentioning

confidence: 99%