Results 1  10
of
67
A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra
, 1990
"... We present a new piv otbased algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following prope ..."
Abstract

Cited by 222 (32 self)
 Add to MetaCart
We present a new piv otbased algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties: (a) No additional storage is required beyond the input data; (b) The output list produced is free of duplicates; (c) The algorithm is extremely simple, requires no data structures, and handles all degenerate cases; (d) The running time is output sensitive for nondegenerate inputs; (e) The algorithm is easy to efficiently parallelize. For example, the algorithm finds the v vertices of a polyhedron in R d defined by a nondegenerate system of n inequalities (or dually, the v facets of the convex hull of n points in R d, where each facet contains exactly d given points) in time O(ndv) and O(nd) space. The v vertices in a simple arrangement of n hyperplanes in R d can be found in O(n²dv) time and O(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.
Algorithms for Sequential Decision Making
, 1996
"... Sequential decision making is a fundamental task faced by any intelligent agent in an extended interaction with its environment; it is the act of answering the question "What should I do now?" In this thesis, I show how to answer this question when "now" is one of a finite set of ..."
Abstract

Cited by 212 (8 self)
 Add to MetaCart
Sequential decision making is a fundamental task faced by any intelligent agent in an extended interaction with its environment; it is the act of answering the question "What should I do now?" In this thesis, I show how to answer this question when "now" is one of a finite set of states, "do" is one of a finite set of actions, "should" is maximize a longrun measure of reward, and "I" is an automated planning or learning system (agent). In particular,
On the complexity of solving Markov decision problems
 IN PROC. OF THE ELEVENTH INTERNATIONAL CONFERENCE ON UNCERTAINTY IN ARTIFICIAL INTELLIGENCE
, 1995
"... Markov decision problems (MDPs) provide the foundations for a number of problems of interest to AI researchers studying automated planning and reinforcement learning. In this paper, we summarize results regarding the complexity of solving MDPs and the running time of MDP solution algorithms. We argu ..."
Abstract

Cited by 159 (12 self)
 Add to MetaCart
Markov decision problems (MDPs) provide the foundations for a number of problems of interest to AI researchers studying automated planning and reinforcement learning. In this paper, we summarize results regarding the complexity of solving MDPs and the running time of MDP solution algorithms. We argue that, although MDPs can be solved efficiently in theory, more study is needed to reveal practical algorithms for solving large problems quickly. To encourage future research, we sketch some alternative methods of analysis that rely on the structure of MDPs.
lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm
 POLYTOPES – COMBINATORICS AND COMPUTATION
, 2000
"... This paper describes an improved implementation of the reverse search vertex enumeration/convex hull algorithm for ddimensional convex polyhedra. The implementation uses a lexicographic ratio test to resolve degeneracy, works on bounded or unbounded polyhedra and uses exact arithmetic with all int ..."
Abstract

Cited by 72 (4 self)
 Add to MetaCart
(Show Context)
This paper describes an improved implementation of the reverse search vertex enumeration/convex hull algorithm for ddimensional convex polyhedra. The implementation uses a lexicographic ratio test to resolve degeneracy, works on bounded or unbounded polyhedra and uses exact arithmetic with all integer pivoting. It can also be used to compute the volume of the convex hull of a set of points. For a polyhedron with m inequalities in d variables and known extreme point, it finds all bases in time O(md2) per basis. This implementation can handle problems of quite large size, especially for simple polyhedra (where each basis corresponds to a vertex and the complexity reduces to O(md) per vertex). Computational experience is included in the paper, including a comparison with an earlier implementation.
Deformed Products and Maximal Shadows of Polytopes
 ADVANCES IN DISCRETE AND COMPUTATIONAL GEOMETRY, AMER. MATH. SOC., PROVIDENCE, CONTEMPORARY MATHEMATICS 223
, 1996
"... We present a construction of deformed products of polytopes that has as special cases all the known constructions of linear programs with "many pivots," starting with the famous KleeMinty cubes from 1972. Thus we obtain sharp estimates for the following geometric quantities for ddimensio ..."
Abstract

Cited by 36 (1 self)
 Add to MetaCart
(Show Context)
We present a construction of deformed products of polytopes that has as special cases all the known constructions of linear programs with "many pivots," starting with the famous KleeMinty cubes from 1972. Thus we obtain sharp estimates for the following geometric quantities for ddimensional simple polytopes with at most n facets: ffl the maximal number of vertices on an increasing path, ffl the maximal number of vertices on a "greedy" greatest increase path, and ffl the maximal number of vertices of a 2dimensional projection. This, equivalently, provides good estimates for the worstcase behaviour of the simplex algorithm on linear programs with these parameters with the worstpossible, the greatest increase, and the shadow vertex pivot rules. The bounds on the maximal number of vertices on an increasing path or a greatest increase path unify and slightly improve a number of known results. One bound on the maximal number of vertices of a 2dimensional projection is new: we show ...
A PRACTICAL ANTICYCLING PROCEDURE FOR LINEARLY CONSTRAINED OPTIMIZATION
, 1989
"... A procedure is described for preventing cycling in activeset methods for linearly constrained optimization, including the simplex method. The key ideas are a limited acceptance ofinfeasibilities in all variables, and maintenance of a "working" feasibility tolerance that increases over a l ..."
Abstract

Cited by 34 (4 self)
 Add to MetaCart
A procedure is described for preventing cycling in activeset methods for linearly constrained optimization, including the simplex method. The key ideas are a limited acceptance ofinfeasibilities in all variables, and maintenance of a "working" feasibility tolerance that increases over a long sequence of iterations. The additional work per iteration is nominal, and "stalling" cannot occur with exact arithmetic. The method appears to be reliable, based on computational results for the first 53 linear programming problems in the Netlib set.
CrissCross Methods: A Fresh View on Pivot Algorithms
 Mathematical Programming
, 1997
"... this paper is to present mathematical ideas and ..."
An algorithm for largescale quadratic programming
 IMA Journal of Numerical Analysis
, 1991
"... We describe a method for solving largescale general quadratic programming problems. Our method is based upon a compendium of ideas which have their origins in sparse matrix techniques and methods for solving smaller quadratic programs. We include a discussion on resolving degeneracy, on single phas ..."
Abstract

Cited by 22 (8 self)
 Add to MetaCart
We describe a method for solving largescale general quadratic programming problems. Our method is based upon a compendium of ideas which have their origins in sparse matrix techniques and methods for solving smaller quadratic programs. We include a discussion on resolving degeneracy, on single phase methods and on solving parametric problems. Some numerical results are included. 1.
NOTES ON BLAND'S PIVOTING RULE
, 1978
"... Recently R.G. Bland proposed two new rules for pivot selection in the simplex method. These elegant rules arise from Bland's work on oriented matroids; their virtue is that they never lead to cycling. We investigate the efficiency of the first of them. On randomly generated problems with 50 non ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
Recently R.G. Bland proposed two new rules for pivot selection in the simplex method. These elegant rules arise from Bland's work on oriented matroids; their virtue is that they never lead to cycling. We investigate the efficiency of the first of them. On randomly generated problems with 50 nonnegative,variables and 50 additional inequalities, Bland's rule requires about 400 iterations on the average; the corresponding figure for the popular "largest coefficient " rule is only about 100. Comparable behaviour seems to persist even on highly degenerate problems. On the theoretical side, we analyse the performance of Bland's rule on the classical KleeMinty examples: for problems with n nonnegative variables and n additional inequalities, the number of iterations is bounded from below by the nth Fibonacci number.