The scheduling of sparse matrix-vector multiplication on a massively parallel dap computer

An efficient data structure is presented which supports general unstructured sparse matrix-vector multiplications on a Distributed Array of Processors (DAP). This approach seeks to reduce the inter-processor data movements and organises the operations in batches of massively parallel steps by a heuristic scheduling procedure performed on the host computer. The resulting data structure is of particular relevance to iterative schemes for solving linear systems. Performance results for matrices taken from well known Linear Programming (LP) test problems are presented and analysed

Experimental investigations in combining primal dual interior point method and simplex based LP solvers

The use of a primal dual interior point method (PD) based optimizer as a robust linear programming (LP) solver is now well established. Instead of replacing the sparse simplex algorithm (SSX), the PD is increasingly seen as complementing it. The progress of PD iterations is not hindered by the degeneracy or the stalling problem of the SSX, indeed it reaches the 'near optimum' solution very quickly. The SSX algorithm, in contrast, is not affected by the boundary conditions which slow down the convergence of the PD. If the solution to the LP problem is non unique, the PD algorithm converges to an interior point of the solution set while the SSX algorithm finds an extreme point solution. To take advantage of the attractive properties of both the PD and the SSX, we have designed a hybrid framework whereby cross over from PD to SSX can take place at any stage of the PD optimization run. The cross over to SSX involves the partition of the PD solution set to active and dormant variables. In this paper we examine the practical difficulties in partitioning the solution set, we discuss the reliability of predicting the solution set partition before optimality is reached and report the results of combining exact and inexact prediction with SSX basis recovery

Summary Conclusions on Computational Experience and the Explanatory Value of Condition Measures for Linear Optimization*

The modern theory of condition measures for convex optimization problems was initially developed for convex problems in conic format, and several aspects of the theory have now been extended to handle non-conic formats as well. In this theory, the (Renegar-) condition measure C(d) for a problem instance with data d=(A,b,c) has been shown to be connected to bounds on a wide variety of behavioral and computational characteristics of the problem instance, from sizes of optimal solutions to the complexity of algorithms. Herein we test the practical relevance of the condition measure theory, as applied to linear optimization problems that one might typically encounter in practice. Using the NETLIB suite of linear optimization problems as a test bed, we found that 71% of the NETLIB suite problem instances have infinite condition measure. In order to examine condition measures of the problems that are the actual input to a modern IPM solver, we also computed condition measures for the NETLIB suite problems after pre-preprocessing by CPLEX 7.1. Here we found that 19% of the post-processed problem instances in the NETLIB suite have infinite condition measure, and that log C(d) of the post-processed problems is fairly nicely distributed. Furthermore, there is a positive linear relationship between IPM iterations and log C(d) of the post-processed problem instances (significant at the 95% confidence level), and 42% of the variation in IPM iterations among the NETLIB suite problem instances is accounted for by log C(d) of the post-processed problem instances.Singapore-MIT Alliance (SMA

Adapting the interior point method for the solution of linear programs on high performance computers

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

A double-pivot degenerate-robust simplex algorithm for linear programming

A double pivot algorithm that combines features of two recently published papers by these authors is proposed. The proposed algorithm is implemented in MATLAB. The MATLAB code is tested, along with a MATLAB implementation of Dantzig's algorithm, for several test sets, including a set of cycling LP problems, Klee-Minty's problems, randomly generated linear programming (LP) problems, and Netlib benchmark problems. The test result shows that the proposed algorithm is (a) degenerate-tolerance as we expected, and (b) more efficient than Dantzig's algorithm for large size randomly generated LP problems but less efficient for Netlib benchmark problems and small size randomly generated problems in terms of CPU time.Comment: 21 pages, 1 figure, and 2 table