A Nearly Asynchronous Parallel Lp-Based Algorithm for the Convex Hull Problem in Multidimensional Space

Abstract

The convex hull of a set A of n points in ℜm generates a polytope P. The frame F of A is the set of extreme points of P. The frame problem, the identification of F given A, is central to problems in operations research and computer science. In OR it occurs in specialized areas of optimization theory: stochastic programming and redundancy in linear programming. In CS it is an important problem in computational geometry. The problem also appears in economics and statistics. The frame problem is computationally intensive and this limits its applications. The standard LP-based approaches for identifying F solve several linear programs with m rows and n — 1 columns, one for each element of A. In this paper we report on a parallel procedure for identifying F using a new LP-based approach. The new approach also uses linear programs with m rows, but the linear programs which must be solved begin with a small number of columns and grow in size, never exceeding the number of points of F. On a small set of test problems, the serial time to identify F varied from one-half to two-thirds that of an enhanced implementation of the standard approach. We dis cuss parallelization of this algorithm for the MIMD environment. On a suite of test problems, our parallel MIMD nearly asynchronous implementation on the Sequent Symmetry S81 achieved a speedup factor of 7 to 13 using up to 16 processors. These developments will permit the solution of problems previously considered too large.KeywordsData Envelopment AnalysisConvex HullExtreme PointParallel AlgorithmStochastic ProgrammingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.