Multi GPU Implementation of the Simplex Algorithm

Abstract

International audienceThe Simplex algorithm is a well known method to solve linear programming (LP) problems. In this paper, we propose an implementation via CUDA of the Simplex method on a multi GPU architecture. Computational tests have been carried out on randomly generated instances for non-sparse LP problems. The tests show a maximum speedup of 24.5 with two Tesla C2050 boards. I. INTRODUCTION Initially developed for real time and high-definition 3D graphic applications, Graphics Processing Units (GPUs) have gained recently attention for High Performance Computing applications. Indeed, the peak computational capabilities of modern GPUs exceeds the one of top-of-the-line central processing units (CPUs). GPUs are highly parallel, multithreaded, manycore units. In November 2006, NVIDIA introduced, Compute Unified Device Architecture (CUDA), a technology that enables users to solve many complex problems on their GPU cards (see for example [1]-[4]). Some related works have been presented on the parallel implementation of algorithms on GPU for linear programming (LP) problems. O'Leary and Jung have proposed in [5] a combined CPU-GPU implementation of the Interior Point Method for LP; computational results carried out on NETLIB LP problems [6] for at most 516 variables and 758 constraints, show that some speedup can be obtained by using GPU for sufficiently large dense problems. Spampinato and Elster have proposed in [7] a parallel implementation of the revised Simplex method for LP on GPU with NVIDIA CUBLAS [8] and NVIDIA LAPACK [9] libraries. Tests were carried out on randomly generated LP problems of at most 2000 variables and 2000 constraints. The implementation showed a maximum speedup of 2.5 on a NVIDIA GTX 280 GPU as compared with sequential implementation on CPU with Intel Core2 Quad 2.83 GHz. Bieling, Peschlow and Martini have proposed in [10] an other implementation of the revised Simplex method on GPU. This implementation permits one to speed up solution with a maximum factor of 18 in single precision on a NVIDIA GeForce 9600 GT GPU card as compared with GLPK solver run on Intel Core 2 Duo 3GHz CPU. In [11], we have presented a parallel implementation via CUD

Simplex Parallelization in a Fully Hybrid Hardware Platform

“…Experiments carried out on an Intel Xeon 3GHz and a GTX 260 GPU have shown substantial speedup of 12.5 in double precision, for randomly generated LP problems of size up to 8000x8000. The authors have also extended their work on a multi-GPU implementation [33] and their computational results showed a maximum speedup of 24.5, using two Tesla C2050 boards. Meyer et al [34] proposed a mono-and a multi-GPU implementation of the tableau simplex algorithm, and they compared its performance to the serial Clp solver, using a Tesla S1070 board with T10 GPUs.…”

“…Later on we present the additional performance gains achieved by our CPU/GPU collaboration (OpenMP+CUDA) scheme over the GPU-only approach, thus demonstaring the really high level of improvements that can be offered by the use of a combined CPU/GPU computing approach in hybrid (multi-node, multi-core) environments that involve CUDA-enabled GPUs too. The measurements presented in Tables III and IV have been taken over a dense randomly generated LP problem of size equal to 10000x10000 and similar properties as in [19,33].…”

“…They are also comparable to other relevant approaches in the literature, and quite better than the ones of [20] Additionally, in Fig.1 the behavior of our GPU-only approach over different sizes of LP problems is shown, in terms of the corresponding speedup curves). The experiments have been performed over randomly generated dense LP problems ranging in size from 640x640 to 10000x10000, of similar properties as in [19,33], and with double precision arithmetic. Note also that the speedup values have been computed (without loss of generality) comparing to the 1-core/sequential implementation.…”

GPU computing in discrete optimization. Part II: Survey focused on routing problems

“…They use randomly generated non-sparse LP instances. Also in 2011, the same authors report from a CUDA implementation of the simplex method on a multi GPU architecture [45]. Computational tests on random, non-sparse instances show a maximum speedup of 24.5 with two Tesla C2050.…”

GPU Implementation of the Branch and Bound Method for Knapsack Problems

“…We note that we have also proposed parallel Simplex methods that run on a single GPU or several GPUs (see [14] and [15]). These codes are particularly interesting when one wants for example to compute bounds of knapsack problems.…”