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35
A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra
, 1992
"... 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 ..."
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Cited by 184 (28 self)
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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) Virtually 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(n 2 dv) 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 states, "do" is one ..."
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Cited by 175 (8 self)
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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 ..."
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Cited by 131 (10 self)
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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
, 1998
"... 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 inte ..."
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Cited by 59 (3 self)
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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 ddimensional simpl ..."
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Cited by 30 (1 self)
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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 ...
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 nonnegat ..."
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Cited by 22 (3 self)
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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.
Randomized Simplex Algorithms on KleeMinty Cubes
 COMBINATORICA
, 1994
"... We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes ..."
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Cited by 19 (6 self)
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We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the randomedge simplex algorithm on KleeMinty cubes) conjectured in the literature. At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a KleeMinty cube is exponential when all paths are taken with equal probability.
CrissCross Methods: A Fresh View on Pivot Algorithms
 Mathematical Programming
, 1997
"... this paper is to present mathematical ideas and ..."
Some Generalizations Of The CrissCross Method For Quadratic Programming
 MATH. OPER. UND STAT. SER. OPTIMIZATION
, 1992
"... Three generalizations of the crisscross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. A finite criss ..."
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Cited by 13 (8 self)
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Three generalizations of the crisscross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. A finite crisscross method, based on leastindex resolution, is constructed for solving the LCP. In proving finiteness, orthogonality properties of pivot tableaus and positive semidefiniteness of quadratic matrices are used. In the last section some special cases and two further variants of the quadratic crisscross method are discussed. If the matrix of the LCP has full rank, then a surprisingly simple algorithm follows, which coincides with Murty's `Bard type schema' in the P matrix case.