New variants of bundle methods

Lemaréchal, Claude; Nemirovski, Arkadi; Nesterov, Yurii

doi:10.1007/bf01585555

Cited by 323 publications

(201 citation statements)

References 12 publications

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“…The bundle method was first proposed by Lemarechal [37], and later on further investigated and refined by several authors, e.g., [33,50,38]. It has been developed for finding the approximate minimizer of a non-smooth convex function f (γ) over γ ∈ R n .…”

Section: Bundle Methodsmentioning

confidence: 99%

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Anjos

Ghaddar²,

Hupp

et al. 2013

Facets of Combinatorial Optimization

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This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching. We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k = 3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solv- ing the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far.

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Section: Bundle Methodsmentioning

confidence: 99%

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Anjos

Ghaddar²,

Hupp

et al. 2013

Facets of Combinatorial Optimization

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“…, 2m}. To generate ζ = [ζ 11 ; ζ 12 ; ζ 2 ] ∼ P z , we draw at random index ı from the distribution z 11 and make [ζ 11 ] ı = 1 the only nonzero entry in ζ 11 . ζ 2 is built similarly, with z 2 in the role of z 11 , and ζ 12 is nothing but z 12 .…”

Section: Application Example: 1 Minimization With P Fitmentioning

confidence: 99%

“…Here we explain how a GBSP problem (11)- (12) can be reduced to a "small series" of BSP problems; the strategy to follow originates from [11]. From now on we assume, in addition to A.1-2, that we have an a priori upper boundρ on the optimal value ρ * of (12).…”

Section: Solving the Generalized Bilinear Saddle Point Problemmentioning

confidence: 99%

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Randomized first order algorithms with applications to ℓ 1-minimization

2012

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In this paper we propose randomized first-order algorithms for solving bilinear saddle points problems. Our developments are motivated by the need for sublinear time algorithms to solve large-scale parametric bilinear saddle point problems where cheap online assessment of the solution quality is crucial. We present the theoretical efficiency estimates of our algorithms and discuss a number of applications, primarily to the problem of 1 minimization arising in sparsity-oriented signal processing. We demonstrate, both theoretically and by numerical examples, that when seeking for medium-accuracy solutions of large-scale 1 minimization problems, our randomized algorithms outperform significantly (and progressively as the sizes of the problem grow) the state-of-the art deterministic methods. Mathematics Subject Classification

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“…Further examples can be found in the excellent surveys in [BeT89,BaB96,CeZ97]. Such methods can also in some cases be interpreted as subgradient methods for the minimization of a non-differentiable convex function over a closed convex set (e.g., [Gof78]), several methods for which also use projections onto level sets of convex functions or surrogate linearized subgradient inequalities (as in "poor man's bundle methods"); see, for example, the level methods in [LNN95,Kiw96a,Kiw96b], references found therein, and [Bra93, pp. 61-78].…”

Section: Euclidean Projectionmentioning

confidence: 99%

A survey on the continuous nonlinear resource allocation problem

Patriksson

2008

European Journal of Operational Research

175

135

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Our problem of interest consists of minimizing a separable, convex and differentiable function over a convex set, defined by bounds on the variables and an explicit constraint described by a separable convex function. Applications are abundant, and vary from equilibrium problems in the engineering and economic sciences, through resource allocation and balancing problems in manufacturing, statistics, military operations research and production and financial economics, to subproblems in algorithms for a variety of more complex optimization models. This paper surveys the history and applications of the problem, as well as algorithmic approaches to its solution. The most common techniques are based on finding the optimal value of the Lagrange multiplier for the explicit constraint, most often through the use of a type of line search procedure. We analyze the most relevant references, especially regarding their originality and numerical findings, summarizing with remarks on possible extensions and future research.

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New variants of bundle methods

Cited by 323 publications

References 12 publications

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Randomized first order algorithms with applications to ℓ 1-minimization

A survey on the continuous nonlinear resource allocation problem

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