On convergence rates of subgradient optimization methods

Goffin, Jean-Louis

doi:10.1007/bf01584346

Cited by 204 publications

(115 citation statements)

References 17 publications

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“…wheref is an estimation of the optimal value f * , and coefficient ρ satisfies 0 < ρ ≤ 2 or according to the convergent series method (see Shor [1968] and Goffin [1977]) with…”

Section: Computational Resultsmentioning

confidence: 99%

A Lagrangian relaxation approach to the edge-weighted clique problem

Hunting

Faigle

Kern

2001

European Journal of Operational Research

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Abstract. The b-clique polytope CP n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QP n as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over QP n n . The problem of optimizing linear functions over CP n b has so far been approached via heuristic combinatorial algorithms and cutting-plane methods.We study the structure of CP n b in further detail and present a new computational approach to the linear optimization problem based on Lucena's suggestion of integrating cutting planes into a Lagrangian relaxation of an integer programming problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented.

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“…wheref is an estimation of the optimal value f * , and coefficient ρ satisfies 0 < ρ ≤ 2 or according to the convergent series method (see Shor [1968] and Goffin [1977]) with…”

Section: Computational Resultsmentioning

confidence: 99%

A Lagrangian relaxation approach to the edge-weighted clique problem

Hunting

Faigle

Kern

2001

European Journal of Operational Research

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“…For a non-differentiable objective function like ours, first order subgradient algorithms are often employed. However, the required number of iterations to achieve -close solution is O(1/ 2 ) [9]. Direct application of subgradient algorithms render the optimization problem impractical.…”

Section: A First Order Methodsmentioning

confidence: 99%

Distributed Flow Algorithms for Scalable Similarity Visualization

Quadrianto

Schuurmans

Smola

2010

2010 IEEE International Conference on Data Mining Workshops

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Abstract-We describe simple yet scalable and distributed algorithms for solving the maximum flow problem and its minimum cost flow variant, motivated by problems of interest in objects similarity visualization. We formulate the fundamental problem as a convex-concave saddle point problem. We then show that this problem can be efficiently solved by a first order method or by exploiting faster quasi-Newton steps. Our proposed approach costs at most O(|E|) per iteration for a graph with |E| edges. Further, the number of required iterations can be shown to be independent of number of edges for the first order approximation method. We present experimental results in two applications: mosaic generation and color similarity based image layouting.

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“…Historically, the subgradient methods were the first numerical schemes for non-smooth convex minimization (see [11] and [7] for historical comments). Very soon it was proved that the efficiency estimate of these schemes is of the order 1) where is the desired absolute accuracy of the approximate solution in function value (see also [3]). …”

Section: Introductionmentioning

confidence: 99%

Structural optimization

Soares

Bendsøe

Choi

et al. 2008

Computers & Structures

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Abstract. In this paper we propose a new approach for constructing efficient schemes for non-smooth convex optimization. It is based on a special smoothing technique, which can be applied to functions with explicit max-structure. Our approach can be considered as an alternative to black-box minimization. From the viewpoint of efficiency estimates, we manage to improve the traditional bounds on the number of iterations of the gradient schemes from O 1 2 to O 1 , keeping basically the complexity of each iteration unchanged.

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On convergence rates of subgradient optimization methods

Cited by 204 publications

References 17 publications

A Lagrangian relaxation approach to the edge-weighted clique problem

A Lagrangian relaxation approach to the edge-weighted clique problem

Distributed Flow Algorithms for Scalable Similarity Visualization

Structural optimization

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