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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 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 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 ... [...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 criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts 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. More generally, 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 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 polyhedron. A linear programming algorithm finds a point in the polytope where ... [...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 cube, 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 method, simplex algorithm and the criss-cross algorithm visit all 8 corners in the worst case. In particular, many optimization algorithms for linear programming, 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 matroid, basis-exchange algorithm, exchange pivoting algorithms and also for interior-point algorithms. Description of the cube The Klee–Minty c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Avis
David Michael Avis (born March 20, 1951) is a Canadians, Canadian and United Kingdom, 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 vertex (geometry), vertices of a convex polytope. Selected publications References External links School of Computer Science(McGill Univ.)David Avis’ homepage(McGill Univ.)David Avis' homepage(Kyoto Univ.) 1951 births Living people Researchers in geometric algorithms Stanford University Schoo ... [...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 The Mathematical Optimization Society (MOS), known as the Mathematical Programming Society until 2010, [...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 criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts 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 input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exchange Algorithms
Exchange may refer to: Physics *Gas exchange is the movement of oxygen and carbon dioxide molecules from a region of higher concentration to a region of lower concentration. Places United States * Exchange, Indiana, an unincorporated community * Exchange, Missouri, an unincorporated community * Exchange, Pennsylvania, an unincorporated community * Exchange, West Virginia, an unincorporated community Elsewhere * Exchange Alley, in London, United Kingdom * Exchange District, a historic area in Winnipeg, Manitoba, Canada Business and economy *''Bureau de change'', a business whose customers exchange one currency for another *Cryptocurrency exchange, a business that allows customers to trade cryptocurrencies or digital currencies. *Digital currency exchangers (a.k.a. DCEs or Bitcoin exchanges), businesses that allow customers to trade digital currencies for other assets, such as conventional fiat money, or different digital currencies *Exchange (economics) *Exchange (organized mar ... [...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, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". 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 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 general topological manifolds ofte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
