Optimization problems lie at the core of scientific and engineering endeavors. Solutions to these problems are often compute-intensive. To fulfill their compute-resource requirements, graphics processing unit (GPU) technology is considered a great opportunity. To this end, we focus on linear programming (LP) problem solving on GPUs using revised simplex method (RSM). This method has potentially GPU-friendly tasks, when applied to large dense problems. Basis update (BU) is one such task, which is performed in every iteration to update a matrix called basis-inverse matrix. The contribution of this paper is twofold. Firstly, we experimentally analyzed the performance of existing GPU-based BU techniques. We discovered that the performance of a relatively old technique, in which each GPU thread computed one element of the basis-inverse matrix, could be significantly improved by introducing a vector-copy operation to its implementation with a sophisticated programming framework. Second, we extended the adapted element-wise technique to develop a new BU technique by using three inexpensive vector operations. This allowed us to reduce the number of floating-point operations and conditional processing performed by GPU threads. A comparison of BU techniques implemented in double precision showed that our proposed technique achieved 17.4% and 13.3% average speed-up over its closest competitor for randomly generated and well-known sets of problems, respectively. Furthermore, the new technique successfully updated basisinverse matrix in relatively large problems, which the competitor was unable to update. These results strongly indicate that our proposed BU technique is not only efficient for dense RSM implementations but is also scalable.

Primal-dual interior-point method (PDIPM) is the most efficient technique for solving sparse linear programming (LP) problems. Despite its efficiency, PDIPM remains a compute-intensive algorithm. Fortunately, graphics processing units (GPUs) have the potential to meet this requirement. However, their peculiar architecture entails a positive relationship between problem density and speedup, conversely implying a limited affinity of GPUs for problem sparsity. To overcome this difficulty, the state-of-the-art hybrid (CPU-GPU) implementation of PDIPM exploits presence of supernodes in sparse matrices during factorization. Supernodes are groups of similar columns that can be treated as dense submatrices. Factorization method used in the state-of-the-art solver performs only selected operations related to large supernodes on GPU. This method is known to underutilize GPU's computational power while increasing CPU-GPU communication overhead. These shortcomings encouraged us to adapt another factorization method, which processes sets of related supernodes on GPU, and introduce it to the PDIPM implementation of a popular open-source solver. Our adaptation enabled the factorization method to better mitigate the effects of round-off errors accumulated over multiple iterations of PDIPM. To augment performance gains, we also used an efficient CPU-based matrix multiplication method. When tested for a set of well-known sparse problems, the adapted solver showed average speed-ups of approximately 55X, 1.14X and 1.05X over the open-source solver's original version, the state-of-the-art solver, and a highly optimized proprietary solver known as CPLEX, respectively. These results strongly indicate that our proposed hybrid approach can lead to significant performance gains for solving large sparse problems.

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