Adapting the Interior Point Method for the Solution of Linear Programs on High Performance Computers (original) (raw)
1992, Computer Science and Operations Research
In this paper we describe a unified algorithmic framework for the interior point method (IPM) of solving Linear Programs (LPs) which allows us to adapt it over a range of high performance computer architectures. We set out the reasons as to why IPM makes better use of high performance computer architecture than the sparse simplex method. In the inner iteration of the IPM a search direction is computed using Newton or higher order methods. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system and the design of data structures to take advantage of coarse grain parallel and massively parallel computer architectures are considered in detail. Finally, we present experimental results of solving NETLIB test problems on examples of these architectures and put forward arguments as to why integration of the system within sparse simplex is beneficial. 2. Sparse Simplex and Interior Point Method: Hardware Platforms Progress in the solution of large LPs has been achieved in three ways, namely hardware, software and algorithmic developments. Most of the developments during the 70's and early 80's in the Sparse Simplex method were based on serial computer architecture. The main thrust of these developments were towards exploiting sparsity and finding methods which either reduced simplex iterations or reduced the computational work in each iteration [BIXBY91, M1TAMZ91]. In general these algorithmic and software developments of the sparse simplex method cannot be readily extended to parallel computers. In contrast the interior point methods which have proven to be robust and competitive appear to be better positioned to make use of newly emerging high * The primal-dual algorithm converges to the optimal solution in at most O (n1/2 L) iterations [MONADL89] where n denotes the dimension of the problems and L the input size. It
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