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PrimalDual Methods for Vertex and Facet Enumeration
 Discrete and Computational Geometry
, 1998
"... Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (respectively vertex) to the vertex (respectively halfspace) representation is called vertex enumeration (respectively facet enumeration) ..."
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Cited by 33 (7 self)
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Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (respectively vertex) to the vertex (respectively halfspace) representation is called vertex enumeration (respectively facet enumeration). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper, we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (facet, respectively) enumeration problem is the facet (vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper, we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primaldual ...
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 9 (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...
Linear equations, Inequalities, Linear Programs (LP), and a New Efficient Algorithm
 Pages 136 in Tutorials in OR, INFORMS
, 2006
"... The dawn of mathematical modeling and algebra occurred well over 3000 years ago in several countries (Babylonia, China, India,...). The earliest algebraic systems constructed are systems of linear equations, and soon after, the famous elimination method for solving them was discovered in China and I ..."
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Cited by 7 (7 self)
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The dawn of mathematical modeling and algebra occurred well over 3000 years ago in several countries (Babylonia, China, India,...). The earliest algebraic systems constructed are systems of linear equations, and soon after, the famous elimination method for solving them was discovered in China and India. This effort culminated in the writing of two books that attracted international attention by the Arabic mathematician Muhammad ibnMusa Alkhawarizmi in the firsthalfof9thcentury. The first, AlMaqala fi Hisab aljabr w’almuqabilah (An essay on Algebra and equations), was translated into Latin under the title Ludus Algebrae, the name “algebra ” for the subject came from this Latin title, and Alkhawarizmi is regarded as the father of algebra. Linear algebra is the branch of algebra dealing with systems of linear equations. The second book KitabalJam’awalTafreeqbilHisabalHindiappeared in Latin translation under the title Algoritmi de Numero Indorum (meaning
Greedy Basis Pursuit
, 2006
"... We introduce Greedy Basis Pursuit (GBP), a new algorithm for computing signal representations using overcomplete dictionaries. GBP is rooted in computational geometry and exploits an equivalence between minimizing the ℓ 1norm of the representation coefficients and determining the intersection of th ..."
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Cited by 7 (0 self)
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We introduce Greedy Basis Pursuit (GBP), a new algorithm for computing signal representations using overcomplete dictionaries. GBP is rooted in computational geometry and exploits an equivalence between minimizing the ℓ 1norm of the representation coefficients and determining the intersection of the signal with the convex hull of the dictionary. GBP unifies the different advantages of previous algorithms: like standard approaches to Basis Pursuit, GBP computes representations that have minimum ℓ 1norm; like greedy algorithms such as Matching Pursuit, GBP builds up representations, sequentially selecting atoms. We describe the algorithm, demonstrate its performance, and provide code. Experiments show that GBP can provide a fast alternative to standard linear programming approaches to Basis Pursuit.
NEW LINEAR PROGRAMMING ALGORITHMS, AND SOME OPEN PROBLEMS IN Linear Complementarity
"... Some open research problems in linear complementarity have already been posed among the exercises in previous chapters. Here we discuss some more research problems briefly. ..."
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Some open research problems in linear complementarity have already been posed among the exercises in previous chapters. Here we discuss some more research problems briefly.