A two-stage successive overrelaxation algorithm for solving the symmetric linear complementarity problem

We propose a two-stage successive overrelaxation method (TSOR) algorithm for solving the symmetric linear complementarity problem. After the first SOR preprocessing stage this algorithm concentrates on updating a certain prescribed subset of variables which is determined by exploiting the complementarity property. We demonstrate that this algorithm successfully solves problems with up to ten thousand variables.This material is based on research supported by National Science Foundation Grants DCR-8420963 and DCR-8521228 and Air Force Office of Scientific Research Grants AFSOR-86-0172 and AFSOR-86-0255 while the author was at the Computer Sciences Department at the University of Wisconsin-Madison, USA.     

Parallel gradient projection successive overrelaxation for symmetric linear complementarity problems and linear programs

A gradient projection successive overrelaxation (GP-SOR) algorithm is proposed for the solution of symmetric linear complementary problems and linear programs. A key distinguishing feature of this algorithm is that when appropriately parallelized, the relaxation factor interval (0, 2) isnot reduced. In a previously proposed parallel SOR scheme, the substantially reduced relaxation interval mandated by the coupling terms of the problem often led to slow convergence. The proposed parallel algorithm solves a general linear program by finding its least 2-norm solution. Efficiency of the algorithm is in the 50 to 100 percent range as demonstrated by computational results on the CRYSTAL token-ring multicomputer and the Sequent Balance 21000 multiprocessor.This material is based on research supported by National Science Foundation Grants DCR-8420963 and DCR-8521228 and Air Force Office of Scientific Research Grants AFOSR-86-0172 and AFOSR-86-0255.     

Asynchronous parallel successive overrelaxation for the symmetric linear complementarity problem

Convergence is established for asynchronous parallel successive overrelaxation (SOR) algorithms for the symmetric linear complementarity problem. For the case of a strictly diagonally dominant matrix convergence is achieved for a relaxation factor interval of (0, 2] with line search, and (0, 1] without line search. Computational tests on the Sequent Symmetry S81 multiprocessor give speedup efficiency in the 43%–91% range for the cases for which convergence is established. The tests also show superiority of the asynchronous SOR algorithms over their synchronous counterparts.This material is based on research supported by National Science Foundation Grants DCR-8420963 and DCR-8521228 and Air Force Office of Scientific Research Grant AFOSR-86-0172.     

Multi-sweep asynchronous parallel successive overrelaxation for the nonsymmetric linear complementarity problem

Convergence is established for themulti-sweep asynchronous parallel successive overrelaxation (SOR) algorithm for thenonsymmetric linear complementarity problem. The algorithm was originally introduced in [4] for the symmetric linear complementarity problem. Computational tests show the superiority of the multi-sweep asynchronous SOR algorithm over its single-sweep counterpart on both symmetric and nonsymmetric linear complementarity problems.This material is based on research supported by National Science Foundation Grants CCR-8723091 and DCR-8521228, and Air Force Office of Scientific Research Grants AFOSR-86-0172 and AFOSR-86-0124.     

Interior proximal point algorithm for linear programs

An interior proximal point algorithm for finding a solution of a linear program is presented. The distinguishing feature of this algorithm is the addition of a quadratic proximal term to the linear objective function. This perturbation has allowed us to obtain solutions with better feasibility. Implementation of this algorithm shows that the algorithms. We also establish global convergence and local linear convergence of the algorithm.This research was supported by National Science Foundation Grants DCR-85-21228 and CCR-87-23091 and by Air Force Office of Scientific Research Grants AFOSR-86-0172 and AFOSR-89-0410. It was conducted while the author was a Graduate Student at the Computer Sciences Department, University of Wisconsin, Madison, Wisconsin.     

Finite termination of the proximal point algorithm

This paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm. We give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm.This material is based on research supported by National Science Foundation Grants DCR-8521228 and CCR-8723091, and Air Force Office of Scientific Research Grant AFOSR-86-0172.     

AnO(n logn) algorithm for the voronoi diagram of a set of simple curve segments

LetX be a given set ofn circular and straight line segments in the plane where two segments may interest only at their endpoints. We introduce a new technique that computes the Voronoi diagram ofX inO(n logn) time. This result improves on several previous algorithms for special cases of the problem. The new algorithm is relatively simple, an important factor for the numerous practical applications of the Voronoi diagram.This work was supported by NSF Grants No. DCR-84-01898 and No. DCR-84-01633.     

Parallel successive overrelaxation methods for symmetric linear complementarity problems and linear programs

A parallel successive overrelaxation (SOR) method is proposed for the solution of the fundamental symmetric linear complementarity problem. Convergence is established under a relaxation factor which approaches the classical value of 2 for a loosely coupled problem. The parallel SOR approach is then applied to solve the symmetric linear complementarity problem associated with the least norm solution of a linear program.This work was sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based on research sponsored by National Science Foundation Grant DCR-84-20963 and Air Force Office of Scientific Research Grants AFOSR-ISSA-85-00080 and AFOSR-86-0172.on leave from CRAI, Rende, Cosenza, Italy.     

Finite perturbation of convex programs

This paper concerns a characterization of the finite-perturbation property of a convex program. When this property holds, finite perturbation of the objective function of a convex program leads to a solution of the original problem which minimizes the perturbation function over the set of solutions of the original problem. This generalizes a finite-termination property of the proximal point algorithm for linear programs and characterizes finite Tikhonov regularization of convex programs.This material is based on research supported by National Science Foundation Grants DCR-8521228 and CCR-8723091 and Air Force Office of Scientific Research Grants AFOSR-86-0172 and AFOSR-89-0410.     

The orthogonal convex skull problem

We give a combinatorial definition of the notion of a simple orthogonal polygon beingk-concave, wherek is a nonnegative integer. (A polygon is orthogonal if its edges are only horizontal or vertical.) Under this definition an orthogonal polygon which is 0-concave is convex, that is, it is a rectangle, and one that is 1-concave is orthoconvex in the usual sense, and vice versa. Then we consider the problem of computing an orthoconvex orthogonal polygon of maximal area contained in a simple orthogonal polygon. This is the orthogonal version of the potato peeling problem. AnO(n ²) algorithm is presented, which is a substantial improvement over theO(n ⁷) time algorithm for the general problem.The work of the first author was supported under a Natural Sciences and Engineering Research Council of Canada Grant No. A-5692 and the work of the second author was partially supported by NSF Grants Nos. DCR-84-01898 and DCR-84-01633.     

Iterative linear programming solution of convex programs

An iterative linear programming algorithm for the solution of the convex programming problem is proposed. The algorithm partially solves a sequence of linear programming subproblems whose solution is shown to converge quadratically, superlinearly, or linearly to the solution of the convex program, depending on the accuracy to which the subproblems are solved. The given algorithm is related to inexact Newton methods for the nonlinear complementarity problem. Preliminary results for an implementation of the algorithm are given.This material is based on research supported by the National Science Foundation, Grants DCR-8521228 and CCR-8723091, and by the Air Force Office of Scientific Research, Grant AFOSR-86-0172. The author would like to thank Professor O. L. Mangasarian for stimulating discussions during the preparation of this paper.     

A new approach to the dynamic maintenance of maximal points in a plane

A pointp _i=(x _i, y_i) in thex–y plane ismaximal if there is no pointp _j=(x _j, y_j) such thatx _j>x_i andy _j>y_i. We present a simple data structure, a dynamic contour search tree, which contains all the points in the plane and maintains an embedded linked list of maximal points so thatm maximal points are accessible inO(m) time. Our data structure dynamically maintains the set of points so that insertions takeO(logn) time, a speedup ofO(logn) over previous results, and deletions takeO((logn)²) time.The research of the first author was partially supported by the National Science Foundation under Grant No. DCR-8320214 and by the Office of Naval Research on Contract No. N 00014-86-K-0689. The research of the second author was partially supported by the Office of Naval Research on Contract No. N 00014-86-K-0689.     

Computing the link center of a simple polygon

The link center of a simple polygonP is the set of pointsx insideP at which the maximal link-distance fromx to any other point inP is minimized. Here the link distance between two pointsx, y insideP is defined to be the smallest number of straight edges in a polygonal path insideP connectingx toy. We prove several geometric properties of the link center and present an algorithm that calculates this set in timeO(n ²), wheren is the number of sides ofP. We also give anO(n logn) algorithm for finding an approximate link center, that is, a pointx such that the maximal link distance fromx to any point inP is at most one more than the value attained from the true link center.Work on this paper by the second author has been supported by National Science Foundation Grant DMS-8501947. Work by the third author has been supported by the Canadian National Science and Engineering Research Council, Grant A0332. Work by the fifth author has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation. Work by the seventh author has been supported by a Killam Senior Research Fellowship from the Canada Council, and work by the ninth author has been supported by the National Science Foundation Grants DCR-84-01898 and DCR-84-01633. Part of the work on this paper has been carried out at the Workshop on Movable Separability of Sets at the Bellairs Research Institute of McGill University, Barbados, February 1986. Further acknowledgments can be obtained from the tenth author upon request.     

A parallel algorithm for generalized networks

This paper presents an application of parallel computing techniques to the solution of an important class of planning problems known as generalized networks. Three parallel primal simplex variants for solving generalized network problems are presented. Data structures used in a sequential generalized network code are briefly discussed and their extension to a parallel implementation of one of the primal simplex variants is given. Computational testing of the sequential and parallel codes, both written in Fortran, was done on the CRYSTAL multicomputer at the University of Wisconsin, and the computational results are presented. Maximum efficiency occurred for multiperiod generalized network problems where a speedup approximately linear in the number of processors was achieved.This research was supported in part by NSF grants DCR-8503148 and CCR-8709952 and by AFOSR grant AFOSR-86-0194.     

An exact algorithm for kinodynamic planning in the plane

Planning time-optimal motions has been a major focus of research in robotics. In this paper we consider the following problem: given an object in two-dimensional physical space, an initial point, and a final point, plan a time-optimal obstacle-avoiding motion for this object subject to bounds on the velocity and acceleration of the object. We give the first algorithm which solves the problem exactly in the case where the velocity and acceleration bounds are given in theL _∞ norm. We further prove the following important results: a tracking lemma and a loop-elimination theorem, both of which are applicable to the case of arbitrary norms. The latter result implies that, with or without obstacles, a path which intersects itself can be replaced by one which does not do so and which takes time less than or equal to that taken by the original path. The work of J. Canny and A. Rege was supported by NSF Grants IRI-89-58577 and IRI-90-14490 and by a David and Lucile Packard Fellowship. J. Reif's work was supported in part by DARPA/ARO Contract DAAL03-88-K-0185, Air Force Contract AFSOR-87-0386, ONR Contract N00014-K-0310, and DARPA/ISTO Contract N00014-88-K-0458.     

On 3-pushdown graphs with large separators

For an integers letl ^s (n), thes-iterated logarithm function, be defined inductively:l ⁰ (n)=n,l ^s+1 (n)=log₂ (l ² (n)) fors0. We show that for every fixeds and alln large enough, there is ann-vertex 3-pushdown graph whose smallest separator contains at least(n/l ^s (n)) vertices.The work of the first author was supported in part by NSF Grants MCS-83-03139, DCR-85-11713 and CCR-86-05353.The work of the second author was supported in part by NSF Grants MCS-84-16190.     

Heuristic Procedures for Solving the Discrete Ordered Median Problem

We present two heuristic methods for solving the Discrete Ordered Median Problem (DOMP), for which no such approaches have been developed so far. The DOMP generalizes classical discrete facility location problems, such as the p-median and p-center. The first procedure proposed in this paper is based on a genetic algorithm developed by Moreno Vega (1996) for p-median and p-center problems. Additionally, a second heuristic approach based on the Variable Neighborhood Search metaheuristic (VNS) proposed by Hansen and Mladenović (1997) for the p-median problem is described. An extensive numerical study is presented to show the efficiency of both heuristics and compare them.     

On the power of white pebbles

We construct a family (G _p |p) of directed acyclic graphs such that any black pebble strategy forG _p requiresp ² pebbles whereas 3p+1 pebbles are sufficient when white pebbles are allowed.Supported by the National Science Foundation under contract no. DCR-8407256 and by the office of Naval Research under contract no. N0014-80-0517.     

Minimum principle sufficiency

We characterize the property of obtaining a solution to a convex program by minimizing over the feasible region a linearization of the objective function at any of its solution points (Minimum Principle Sufficiency). For the case of a monotone linear complementarity problem this MPS property is completely equivalent to the existence of a nondegenerate solution to the problem. For the case of a convex quadratic program, the MPS property is equivalent to the span of the Hessian of the objective function being contained in the normal cone to the feasible region at any solution point, plus the cone generated by the gradient of the objective function at any solution point. This in turn is equivalent to the quadratic program having a weak sharp minimum. An important application of the MPS property is that minimizing on the feasible region a linearization of the objective function at a point in a neighborhood of a solution point gives an exact solution of the convex program. This leads to finite termination of convergent algorithms that periodically minimize such a linearization.This material is based on research supported by National Science Foundation Grants DCR-8521228 and CCR-8723091, and Air Force Office of Scientific Research Grants AFOSR 86-0172 and AFOSR and AFOSR 89-0410.     

On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method

Interest in linear programming has been intensified recently by Karmarkar’s publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrier-function methods for nonlinear programming based on applying a logarithmic transformation to inequality constraints. For the special case of linear programming, the transformed problem can be solved by a “projected Newton barrier” method. This method is shown to be equivalent to Karmarkar’s projective method for a particular choice of the barrier parameter. We then present details of a specific barrier algorithm and its practical implementation. Numerical results are given for several non-trivial test problems, and the implications for future developments in linear programming are discussed. The research of the Stanford authors was supported by the U.S. Department of Energy Contract DE-AA03-76SF00326, PA No. DE-AS03-76ER72018; National Science Foundation Grants DCR-8413211 and ECS-8312142; the Office of Naval Research Contract N00014-85-K-0343; and the U.S. Army Research Office Contract DAAG29-84-K-0156. The research of J.A. Tomlin was supported by Ketron, Inc. and the Office of Naval Research Contract N00014-85-C-0338.