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Cunningham's Rule
In mathematical optimization, Cunningham's rule (also known as least recently considered rule or round-robin rule) is an algorithmic refinement of the simplex method for linear programming, linear optimization. The rule was proposed 1979 by W. H. Cunningham to defeat the deformed hypercube constructions by Klee and Minty et al. (see, e.g. Klee–Minty cube). Cunningham's rule assigns a cyclic order to the variables and remembers the last variable to enter the basis. The next entering variable is chosen to be the first allowable candidate starting from the last chosen variable and following the given circular order. History-based rules defeat the deformed hypercube constructions because they tend to average out how many times a variable pivots. It has recently been shown by David Avis and Oliver Friedmann that there is a family of linear programs on which the simplex algorithm equipped with Cunningham's rule requires exponential time. Notes {{Reflist Optimization algorith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex Method
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial ''cones'', and these become proper simplices with an additional constraint. The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function. History George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator. During 1946, his colleague challenged him to mechanize the planning process to distract him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of Wassily Leontief, however, at that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klee–Minty Cube
The Klee–Minty cube or Klee–Minty polytope (named after Victor Klee and George J. Minty) is a unit hypercube of variable dimension whose corners have been perturbed. Klee and Minty demonstrated that George Dantzig's simplex algorithm has poor worst-case performance when initialized at one corner of their "squashed cube". On the three-dimensional version, the simplex algorithm and the criss-cross algorithm visit all 8 corners in the worst case. In particular, many optimization algorithms for linear optimization exhibit poor performance when applied to the Klee–Minty cube. In 1973 Klee and Minty showed that Dantzig's simplex algorithm was not a polynomial-time algorithm when applied to their cube. Later, modifications of the Klee–Minty cube have shown poor behavior both for other basis-exchange pivoting algorithms and also for interior-point algorithms. Description The Klee–Minty cube was originally specified with a parameterized system of linear inequalities, with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Operations Research
''Mathematics of Operations Research'' is a quarterly peer-reviewed scientific journal established in February 1976. It focuses on areas of mathematics relevant to the field of operations research such as continuous optimization, discrete optimization, game theory, machine learning, simulation methodology, and stochastic models. The journal is published by INFORMS (Institute for Operations Research and the Management Sciences). the journal has a 2017 impact factor of 1.078. History The journal was established in 1976. The founding editor-in-chief was Arthur F. Veinott Jr. (Stanford University). He served until 1980, when the position was taken over by Stephen M. Robinson, who held the position until 1986. Erhan Cinlar served from 1987 to 1992, and was followed by Jan Karel Lenstra (1993-1998). Next was Gérard Cornuéjols (1999-2003) and Nimrod Megiddo (2004-2009). Finally came Uri Rothblum (2009-2012), Jim Dai (2012-2018), and the current editor-in-chief Katya Scheinberg (20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Avis
David Michael Avis (born March 20, 1951) is a Canadian and British computer scientist known for his contributions to geometric computations. Avis is a professor in computational geometry and applied mathematics in the School of Computer Science, McGill University, in Montreal. Since 2010, he belongs to Department of Communications and Computer Engineering, School of Informatics, Kyoto University. Avis received his Ph.D. in 1977 from Stanford University. He has published more than 70 journal papers and articles. Writing with Komei Fukuda, Avis proposed a reverse-search algorithm for the vertex enumeration problem; their algorithm generates all of the vertices of a convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo .... Selected publications References Externa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oliver Friedmann
Oliver Friedmann is a German computer scientist and mathematician known for his work on parity games and the simplex algorithm. Friedmann earned his doctorate's degree from the Ludwig Maximilian University of Munich in 2011 under the supervision of Martin Hofmann and Martin Lange. Awards He won the Kleene Award for showing that state-of-the-art policy iteration algorithms for parity games require exponential time in the worst case. He and his coauthors extended the proof techniques to the simplex algorithm and to policy iteration for Markov decision processes. His seminal body of work on lower bounds in convex optimization, leading to a sub-exponential lower bound for Zadeh's rule, was awarded with the Tucker Prize The Tucker Prize for outstanding theses in the area of optimization is sponsored by the Mathematical Optimization Society (MOS). Up to three finalists are presented at each (triennial) International Symposium of the MOS. The winner will receive an .... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Algorithms And Methods
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Optimization problems can be divided into two categ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exchange Algorithms
Exchange or exchanged may refer to: Arts, entertainment and media Film and television * Exchange (film), or ''Deep Trap'', 2015 South Korean psychological thriller * Exchanged (film), 2019 Peruvian fantasy comedy * Exchange (TV program), 2021 South Korean dating reality show Gaming * Exchange (chess), closely related captures of pieces of both players in chess ** The exchange (chess), an exchange of a minor piece for a rook Music * Exchange, a new-age jazz band of Steve Sexton and Gerald O'Brien, and their 1992 self-titled album * ''Exchange'' (EP), by Against All Authority and The Criminals, 1999 * "Exchange" (song), by Bryson Tiller, 2015 * "Exchange" and "(Exchange)", songs on Massive Attack's 1998 album ''Mezzanine'' (album) Business and economics * Bureau de change, or currency exchange * Cryptocurrency exchange, to trade cryptocurrencies or digital currencies * Exchange (economics), in a market economy * Exchange (organized market), where securities etc are bough ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oriented Matroids
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, Surface (topology), surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loop (topology), loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
