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12
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 ..."
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Cited by 36 (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 ...
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 20 (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.
Complexity results for InfiniteHorizon Markov Decision Processes
, 2000
"... Markov decision processes (MDPs) are models of dynamic decision making under uncertainty. These models arise in diverse applications and have been developed extensively in fields such as operations research, control engineering, and the decision sciences in general. Recent research, especially in a ..."
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Cited by 18 (3 self)
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Markov decision processes (MDPs) are models of dynamic decision making under uncertainty. These models arise in diverse applications and have been developed extensively in fields such as operations research, control engineering, and the decision sciences in general. Recent research, especially in artificial intelligence, has highlighted the significance of studying the computational properties of MDP problems. We address
The random facet simplex algorithm on combinatorial cubes
 Random Structures & Algorithms
, 2001
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A Survey on Pivot Rules for Linear Programming
 ANNALS OF OPERATIONS RESEARCH. (SUBMITTED
, 1991
"... The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. Th ..."
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Cited by 11 (1 self)
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The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...
Worstcase Analysis of Strategy Iteration and the Simplex Method
, 2012
"... In this dissertation we study strategy iteration (also known as policy iteration) algorithms for solving Markov decision processes (MDPs) and twoplayer turnbased stochastic games (2TBSGs). MDPs provide a mathematical model for sequential decision making under uncertainty. They are widely used to m ..."
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Cited by 6 (1 self)
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In this dissertation we study strategy iteration (also known as policy iteration) algorithms for solving Markov decision processes (MDPs) and twoplayer turnbased stochastic games (2TBSGs). MDPs provide a mathematical model for sequential decision making under uncertainty. They are widely used to model stochastic optimization problems in various areas ranging from operations research, machine learning, artificial intelligence, economics and game theory. The class of twoplayer turnbased stochastic games is a natural generalization of Markov decision processes that is obtained by introducing an adversary. 2TBSGs form an intriguing class of games whose status in many ways resembles that of linear programming 40 years ago. They can be solved efficiently with strategy iteration algorithms, resembling the simplex method for linear programming, but no polynomial time algorithm is known. Linear programming is an exceedingly important problem with numerous applications. The simplex method was introduced by
A subexponential lower bound for the Least Recently Considered rule for solving linear programs and games
"... The simplex algorithm is among the most widely used algorithms for solving linear programs in practice. Most pivoting rules are known, however, to need an exponential number of steps to solve some linear programs. No nonpolynomial lower bounds were known, prior to this work, for Cunningham’s Least ..."
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The simplex algorithm is among the most widely used algorithms for solving linear programs in practice. Most pivoting rules are known, however, to need an exponential number of steps to solve some linear programs. No nonpolynomial lower bounds were known, prior to this work, for Cunningham’s Least Recently Considered rule [5], which belongs to the family of historybased rules. Also known as the ROUNDROBIN rule, Cunningham’s pivoting method fixes an initial ordering on all variables first, and then selects the improving variables in a roundrobin fashion. We provide the first subexponential (i.e., of the form 2 Ω( √ n)) lower bound for this rule in a concrete setting. Our lower bound is obtained by utilizing connections between pivoting steps performed by simplexbased algorithms and improving switches performed by policy iteration algorithms for 1player and 2player games. We start by building 2player parity games (PGs) on which the policy iteration with the ROUNDROBIN rule performs a subexponential number of iterations. We then transform the parity games into 1player Markov Decision Processes (MDPs) which correspond almost immediately to concrete linear programs. 1
OPTIMA Mathematical Optimization Society Newsletter
, 2011
"... with goodies central to our field. After the summer months most of us are now back to our more usual occupations and our research activities in optimization. I truly hope that you share my anticipation of its moments of collaborative inspiration. One thing is sure: after the successful midyear meet ..."
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with goodies central to our field. After the summer months most of us are now back to our more usual occupations and our research activities in optimization. I truly hope that you share my anticipation of its moments of collaborative inspiration. One thing is sure: after the successful midyear meetings, we are now heading towards the high point of 2012: the ISMP in Berlin. I hear from good sources that preparations are progressing well, and that all augurs are favourable. As you all know, several prizes will be awarded at the ISMP opening ceremony, recognizing the contributions or both younger and more senior colleagues. You undoubtedly have seen the various calls for nominations for the Dantzig, Lagrange, Fulkerson, BealeOrchardHays and Tucker prizes as well as that for the Paul Tseng lectureship. I encourage you to seriously consider nominating one or more optimization researchers for these prizes. These awards and the high scientific standards of their recipients not only recognize