Abstract
This paper introduces the design and implementation of two parallel dual simplex solvers for general large scale sparse linear programming problems. One approach, called PAMI, extends a relatively unknown pivoting strategy called suboptimization and exploits parallelism across multiple iterations. The other, called SIP, exploits purely single iteration parallelism by overlapping computational components when possible. Computational results show that the performance of PAMI is superior to that of the leading open-source simplex solver, and that SIP complements PAMI in achieving speedup when PAMI results in slowdown. One of the authors has implemented the techniques underlying PAMI within the FICO Xpress simplex solver and this paper presents computational results demonstrating their value. In developing the first parallel revised simplex solver of general utility, this work represents a significant achievement in computational optimization.
Highlights
Linear programming (LP) has been used widely and successfully in many practical areas since the introduction of the simplex method in the 1950s
This paper introduces the design and implementation of two parallel dual simplex solvers for general large scale sparse linear programming problems
– Following suboptimization, the collective Forrest–Tomlin (CFT) update [16] updates Bk−1 to Bk−+1t directly, using partial results obtained with Bk−1 which are required for simplex iterations
Summary
Linear programming (LP) has been used widely and successfully in many practical areas since the introduction of the simplex method in the 1950s. Since scalable speedup for general large sparse LP problems appears unachievable, the revised simplex method has been considered unsuitable for parallelisation Since it corresponds to the computationally efficient serial technique, any improvement in performance due to exploiting parallelism in the revised simplex method is a worthwhile goal. Dual simplex implementations are generally preferred, almost all the work by others on parallel simplex has been restricted to the primal algorithm, the only published work on dual simplex parallelisation known to the authors being due to Bixby and Martin [1] It appeared in the early 2000s, their implementation included neither the BFRT nor hyper-sparse linear system solution techniques so there is immediate scope to extend their work.
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