Parallel search paths for the simplex algorithm
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It is well known that the simplex method is inherently a sequential algorithm with little scope for parallelization. Even so, during the last decades several attempts were made to parallelize it since it is one of the most important algorithms for solving linear optimization problems. Such parallelization ideas mostly rely on iteration parallelism and overlapping. Since the simplex method goes through a series of basic solutions until it finds an optimal solution, each of them must be available before performing the next basis change. This phenomenon imposes a limit on the performance of the parallelized version of the simplex method which uses overlapping iterations. Another approach can be considered if we think about alternative paths on the n-dimensional simplex polyhedron. As the simplex method goes through the edges of this polyhedron it is generally true that the speed of convergence of the algorithm is not smooth. It depends on the actual part of the surface. If a parallel version of the simplex algorithm simultaneously goes on different paths on this surface a highly reliable algorithm can be constructed. There is no known dominating strategy for pivot selection. Therefore, one can try different pivot selection methods in parallel in order to guide the algorithm on different pathways. This approach can be used effectively with periodic synchronization on shared memory multi-core computing environments to speed up the solution algorithm and get around numerically and/or algorithmically difficult situations throughout the computations.
KeywordsLinear programming Simplex method Parallelization
This publication/research has been supported by the European Union and Hungary and co-financed by the European Social Fund through the Project TÁMOP-4.2.2.C-11/1/KONV-2012-0004—National Research Center for Development and Market Introduction of Advanced Information and Communication Technologies.
- Bieling J, Peschlow P, Martini P (2010) An efficient GPU implementation of the revised simplex method. In: 2010 IEEE international symposium on parallel distributed processing, workshops and PhD Forum (IPDPSW), pp 1–8Google Scholar
- Gay D (1985) Electronic mail distribution of linear programming test problems. Math Program Soc COAL Newsl 13:10–12Google Scholar
- Grama A, Karypia G, Gupta A, Kumar V (2003) Introduction to parallel computing: design and analysis of algorithms. Addison-Wesley, ReadingGoogle Scholar
- Hall J, McKinnon K (1996) PARSMI, a parallel revised simplex algorithm incorporating minor iterations and Devex pricing. In: Waśniewski J, Dongarra J, Madsen K, Olesen D (eds) Applied parallel computing industrial computation and optimization. Lecture notes in computer science, vol 1184. Berlin, Springer, pp 359–368Google Scholar
- Huangfu Q, Hall J (2015b) Parallelizing the dual revised simplex method. Technical report, Cornell UniversityGoogle Scholar
- Koberstein A (2005) The dual simplex method, techniques for a fast and stable implementation. PhD thesis, Universität Paderborn, PaderbornGoogle Scholar
- Lalami M, Boyer V, El-Baz D (2011) Efficient implementation of the simplex method on a CPU-GPU system. In: Proceedings of the 2011 IEEE international symposium on parallel and distributed processing workshops and PhD Forum, IPDPSW’11, Washington, DC, USA. IEEE Computer Society, pp 1999–2006Google Scholar
- Stágel B, Tar P, Maros I (2015) The Pannon Optimizer - a linear programming solver for research purposes. In: Proceedings of the 5th international conference on recent achievements in mechatronics, automation, computer science and robotics, vol 1(1), pp 293–301Google Scholar
- Tar P, Maros I (2012) Product form of the inverse revisited. In: 3rd student conference on operational research, Ravizza S, Holborn P (eds) OpenAccess Series in Informatics (OASIcs), Dagstuhl, Germany. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, vol 22, pp 64–74Google Scholar
- University of Pannonia (2015) Pannon optimizer. http://sourceforge.net/projects/pannonoptimizer/
- Yarmish G (2001) A distributed implementation of the simplex method. PhD thesis, Polytechnic University, Brooklyn, NY, USA. AAI3006399Google Scholar