A parallel interior point decomposition algorithm for block angular semidefinite programs

Sivaramakrishnan, Kartik

doi:10.1007/s10589-008-9187-4

Cited by 15 publications

(12 citation statements)

References 30 publications

(71 reference statements)

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“…A detailed overview is given in Section II. Also, lately, some alternative very promising efforts have been made, based on the block angular structure (or decomposition) of the initially transformed problems [5][6], and they have led to very good results for large scale problems over distributed memory multicore environments.…”

Section: Introductionmentioning

confidence: 99%

Simplex Parallelization in a Fully Hybrid Hardware Platform

Mamalis¹,

Perlitis²

2017

ijacsa

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Abstract-The simplex method has been successfully used in solving linear programming (LP) problems for many years. Parallel approaches have also extensively been studied due to the intensive computations required, especially for the solution of large LP problems. Furthermore, the rapid proliferation of multicore CPU architectures as well as the computational power provided by the massive parallelism of modern GPUs have turned CPU / GPU collaboration models increasingly into focus over the last years for better performance. In this paper, a highly scalable implementation framework of the standard full tableau simplex method is first presented, over a hybrid parallel platform which consists of multiple multicore nodes interconnected via a high-speed communication network. The proposed approach is based on the combined use of MPI and OpenMP, adopting a suitable column-based distribution scheme for the simplex tableau. The parallelization framework is then extended in such a way that it can exploit concurrently the full power of the provided resources on a multicore single-node environment with a CUDA-enabled GPU (i.e. using the CPU cores and the GPU concurrently), based on a suitable hybrid multithreading/GPU offloading scheme with OpenMP and CUDA. The corresponding experimental results show that the hybrid MPI+OpenMP based parallelization scheme leads to particularly high speed-up and efficiency values, considerably better than in other competitive approaches, and scaling well even for very large / huge linear problems. Furthermore, the performance of the hybrid multithreading/GPU offloading scheme is clearly superior to both the OpenMP-only and the GPU-only based implementations in almost all cases, which validates the worth of using both resources concurrently. The most important, when it is used in combination with MPI in a multi-node (fully hybrid) environment, it leads to substantial improvements in the speedup achieved for large and very large LP problems.

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Section: Introductionmentioning

confidence: 99%

Simplex Parallelization in a Fully Hybrid Hardware Platform

Mamalis¹,

Perlitis²

2017

ijacsa

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“…The next lemma establishes an upper bound on the right hand side of inequality (28). The structure of this proof has some similarities to that of Lemma 9 in [38] for the case of linear programming.…”

Section: Lemma 9 Letmentioning

confidence: 72%

A second-order cone cutting surface method: complexity and application

Oskoorouchi

Mitchell

2007

Comput Optim Appl

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“…Sivaramakrishnan [130] developed a parallel decomposition approach for block-angular semidefinite programming problems. The primal matrix in these problems is comprised of r Tutorials in Operations Research, c 2009 INFORMS smaller diagonal blocks, each with its own set of linear constraints.…”

Section: Developing a Practical Algorithmmentioning

confidence: 99%

“…The solutions X i to the subproblems lead to the subgradients −A i (X i ) of Θ i (y). The decomposition algorithm in Sivaramakrishnan [130] is applied to general semidefinite programs by exploiting chordal extensions (de Klerk [32]). In this approach, a graph is constructed for problem (SDP) with nodes corresponding to the rows of the matrix X.…”

Section: R;mentioning

confidence: 99%

Cutting Plane Methods and Subgradient Methods

Mitchell

2009

Decision Technologies and Applications

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Interior point methods have proven very successful at solving linear programming problems. When an explicit linear programming formulation is either not available or is too large to employ directly, a column generation approach can be used. Examples of column generation approaches include cutting plane methods for integer programming and decomposition methods for many classes of optimization problems. This tutorial discusses the use of interior point methods in a column generation scheme. Semidefinite programming relaxations of combinatorial optimization problems are often tighter than linear programming relaxations. This tutorial describes some research in using semidefinite programming relaxations to find exact solutions to combinatorial optimization problems. Semidefinite programs are expensive to solve directly, so cutting surface approaches to solving them are also considered. Finally, this tutorial looks at recent smoothing techniques for solving nonsmooth optimization problems using a subgradient approach. These methods have some links to cutting surface approaches.

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A parallel interior point decomposition algorithm for block angular semidefinite programs

Abstract: Block angular semidefinite programs, Matrix completion, Decomposition and nonsmooth optimization, Interior point methods, Parallel optimization,

Cited by 15 publications

References 30 publications

Simplex Parallelization in a Fully Hybrid Hardware Platform

Simplex Parallelization in a Fully Hybrid Hardware Platform

A second-order cone cutting surface method: complexity and application

Cutting Plane Methods and Subgradient Methods

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