Implementation of a primal–dual method for SDP on a shared memory parallel architecture

Borchers, Brian; Young, Joseph G.

doi:10.1007/s10589-007-9030-3

Cited by 49 publications

(55 citation statements)

References 17 publications

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“…We do this by using a Gaussian kernel W D = exp(− 1 2σ 2 D 2 ) and set the width parameter σ =D/ √ 2, which gives good results in practice. Another almost equally good choice is σ = 1 3 , which can be justified by the fact that the Google distance is scale invariant and mostly in [0,1]. Table 1 shows the number of clustering errors, i.e., the number of data points that are sorted to a different group than the intended one, respectively, on the data sets just described.…”

Section: Experimental Results With the Google Distancementioning

confidence: 99%

“…Efficient solvers are available for this kind of optimization problems, such as CSDP [1] or SeDuMi [10]. In order to implement the constraints Y ij ≥ − 1 k−1 with these solvers, positive slack variables Z ij have to be introduced together with the equality constraints Y ij − Z ij = − 1 k−1 .…”

Section: Max-k-cutmentioning

confidence: 99%

“…The respective complexity is O(n 3 + m 3 ) (see [1]), where m is the number of constraints. If k = 2, then m = n and the overall complexity is cubic.…”

Section: Max-k-cut Versus Norm-k-cutmentioning

confidence: 99%

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Clustering Pairwise Distances with Missing Data: Maximum Cuts Versus Normalized Cuts

Poland

Zeugmann

2006

Lecture Notes in Computer Science

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Abstract. Clustering algorithms based on a matrix of pairwise similarities (kernel matrix) for the data are widely known and used, a particularly popular class being spectral clustering algorithms. In contrast, algorithms working with the pairwise distance matrix have been studied rarely for clustering. This is surprising, as in many applications, distances are directly given, and computing similarities involves another step that is error-prone, since the kernel has to be chosen appropriately, albeit computationally cheap. This paper proposes a clustering algorithm based on the SDP relaxation of the max-k-cut of the graph of pairwise distances, based on the work of Frieze and Jerrum. We compare the algorithm with Yu and Shi's algorithm based on spectral relaxation of a norm-k-cut. Moreover, we propose a simple heuristic for dealing with missing data, i.e., the case where some of the pairwise distances or similarities are not known. We evaluate the algorithms on the task of clustering natural language terms with the Google distance, a semantic distance recently introduced by Cilibrasi and Vitányi, using relative frequency counts from WWW queries and based on the theory of Kolmogorov complexity.

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Section: Experimental Results With the Google Distancementioning

confidence: 99%

Section: Max-k-cutmentioning

confidence: 99%

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Clustering Pairwise Distances with Missing Data: Maximum Cuts Versus Normalized Cuts

Poland

Zeugmann

2006

Lecture Notes in Computer Science

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“…License: open source Developer: Brian Borchers Capabilities: LP, SDP Algorithm: Interior-point method Special features: Callable library, Interface to R Website: http://projects.coin-or.org/Csdp Reference: [4,20] Parallel version in both shared [21] and distributed [22] memory configurations are available. CSDP is one of the most advanced solvers, it exploits sparsity in the data matrices A (i) .…”

Section: Csdpmentioning

confidence: 99%

Conic Optimization Software

Pólik

2011

Wiley Encyclopedia of Operations Research and Management Science

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“…As a result, parallel versions of several SDP software packages have been developed like PDSDP [1], SDPARA [27] and a parallel version of CSDP [5]. The first two are designed for PC clusters using MPI 1 and ScaLAPACK 2 and the last one is designed for a shared memory computer architecture [5].…”

Section: Introductionmentioning

confidence: 99%

Parallel implementation of a semidefinite programming solver based on CSDP on a distributed memory cluster

Ivanov

Klerk

2010

Optimization Methods and Software

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General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal Take down policyIf you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. AbstractIn this paper we present the algorithmic framework and practical aspects of implementing a parallel version of a primal-dual semidefinite programming solver on a distributed memory computer cluster. Our implementation is based on the CSDP solver and uses a message passing interface (MPI), and the ScaLAPACK library. A new feature is implemented to deal with problems that have rank-one constraint matrices. We show that significant improvement is obtained for a test set of problems with rank one constraint matrices. Moreover, we show that very good parallel efficiency is obtained for large-scale problems where the number of linear equality constraints is very large compared to the block sizes of the positive semidefinite matrix variables.

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Implementation of a primal–dual method for SDP on a shared memory parallel architecture

Cited by 49 publications

References 17 publications

Clustering Pairwise Distances with Missing Data: Maximum Cuts Versus Normalized Cuts

Clustering Pairwise Distances with Missing Data: Maximum Cuts Versus Normalized Cuts

Conic Optimization Software

Parallel implementation of a semidefinite programming solver based on CSDP on a distributed memory cluster

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