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Extension Of A Polyhedron
In convex geometry and polyhedral combinatorics, the extension complexity is a convex polytope P is the smallest number of facets among convex polytopes Q that have P as a projection. In this context, Q is called an extended formulation of P; it may have much higher dimension than P. The extension complexity depends on the precise shape of P, not just on its combinatorial structure. For instance, regular polygons with n sides have extension complexity O(\log n) (expressed using big O notation), but some other convex n-gons have extension complexity at least proportional to \sqrt. If a polytope describing the feasible solutions to a combinatorial optimization problem has low extension complexity, this could potentially be used to devise efficient algorithms for the problem, using linear programming on its extended formulation. For this reason, researchers have studied the extension complexity of the polytopes arising in this way. For instance, it is known that the matching polyt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theory, etc. Classification According to the Mathematics Subject Classification MSC2010, the mathematical discipline ''Convex and Discrete Geometry'' includes three major branches: * general convexity * polytopes and polyhedra * discrete geometry (though only portions of the latter two are included in convex geometry). General convexity is further subdivided as follows: *axiomatic and generalized convexity *convex sets without dimension restrictions *convex sets in topological vector spaces *convex sets in 2 dimensions (including convex curves) *convex sets in 3 dimensions (including convex surfaces) *convex sets in ''n'' dimensions (including convex hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Matroid
In mathematics, a regular matroid is a matroid that can be represented over all fields. Definition A matroid is defined to be a family of subsets of a finite set, satisfying certain axioms. The sets in the family are called "independent sets". One of the ways of constructing a matroid is to select a finite set of vectors in a vector space, and to define a subset of the vectors to be independent in the matroid when it is linearly independent in the vector space. Every family of sets constructed in this way is a matroid, but not every matroid can be constructed in this way, and the vector spaces over different fields lead to different sets of matroids that can be constructed from them. A matroid M is regular when, for every field F, M can be represented by a system of vectors over F.. Properties If a matroid is regular, so is its dual matroid, and so is every one of its minors. Every direct sum of regular matroids remains regular. Every graphic matroid (and every co-graphic matro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optima (journal)
The Mathematical Optimization Society (MOS), known as the Mathematical Programming Society until 2010, . is an international association of researchers active in . The MOS encourages the research, development, and use of optimization—including , [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete & Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * ''Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents''/Engineering, Computing and Technology Notable articles The articles by Gil Kalai with a proof of a subexponential upper bound on the diameter of a polyhedron and by Samuel Ferguson on the Kepler conjecture, both published in Discrete & Computational geometry, earned their author the Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e .... References External link ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Operations Research
''Annals of Operations Research'' is a peer-reviewed academic journal published by Springer Science+Business Media. It was previously published by Baltzer Science Publishers. The journal publishes 24 issues a year that focus on the theoretical, practical, and computational aspects of operations research. It also publishes periodic special volumes focusing on defined fields of operations research. Editors-in-chief The following is a list of persons that have been editor-in-chief of The Annals of Operations Research. * Peter L. Hammer * Endre Boros Abstracting and indexing ''Annals of Operations Research'' is abstracted and indexed in DBLP, Journal Citation Reports, Mathematical Reviews, Research Papers in Economics, SCImago Journal Rank, Scopus, Science Citation Index, Zentralblatt MATH, among others. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Programming
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] |
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 (2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrahedron
In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of positive semidefinite matrices forms a convex cone in , and a spectrahedron is a shape that can be formed by intersecting this cone with a linear affine subspace. Spectrahedra are the feasible regions of semidefinite programs. The images of spectrahedra under linear or affine transformations are called ''projected spectrahedra'' or ''spectrahedral shadows''. Every spectrahedral shadow is a convex set that is also semialgebraic, but the converse (conjectured to be true until 2017) is false. An example of a spectrahedron is the spectraplex, defined as : \mathrm_n = \ where \mathbf^n_+is the set of positive semidefinite matrices and \operatorname(X) is the trace of the matrix X. The spectraplex is a compact set, and can be thought of as the "semidefinite" analog of the simplex. See also * N-ellipse In geometry, the -ellipse is a generalization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semidefinite Programming
Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron. Semidefinite programming is a relatively new field of optimization which is of growing interest for several reasons. Many practical problems in operations research and combinatorial optimization can be modeled or approximated as semidefinite programming problems. In automatic control theory, SDPs are used in the context of linear matrix inequalities. SDPs are in fact a special case of cone programming and can be efficiently solved by interior point methods. All linear programs and (convex) quadratic programs can be expressed as SDPs, and via hierarchies of SDPs the solutions of polynomial optimization problems can be approximated. Semidefinite programming has been use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Polytope
In mathematics, a matroid polytope, also called a matroid basis polytope (or basis matroid polytope) to distinguish it from other polytopes derived from a matroid, is a polytope constructed via the bases of a matroid. Given a matroid M, the matroid polytope P_M is the convex hull of the indicator vectors of the bases of M. Definition Let M be a matroid on n elements. Given a basis B \subseteq \ of M, the indicator vector of B is :\mathbf e_B := \sum_ \mathbf e_i, where \mathbf e_i is the standard ith unit vector in \mathbb^n. The matroid polytope P_M is the convex hull of the set :\ \subseteq \mathbb^n. Examples * Let M be the rank 2 matroid on 4 elements with bases :: \mathcal(M) = \. :That is, all 2-element subsets of \ except \ . The corresponding indicator vectors of \mathcal(M) are :: \. :The matroid polytope of M is : P_M = \text\. :These points form four equilateral triangles at point \, therefore its convex hull is the square pyramid by definition. * Let N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Combinatorics
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex). Additionally, many computer scientists use the phrase “polyhedral combinatorics” to describe research into precise descriptions of the faces of certain specific polytopes (especially 0-1 polytopes, whose vertices are subsets of a hypercube) arising from integer progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matching Polytope
In graph theory, the matching polytope of a given graph is a geometric object representing the possible matchings in the graph. It is a convex polytope each of whose corners corresponds to a matching. It has great theoretical importance in the theory of matching. Preliminaries Incidence vectors and matrices Let ''G'' = (''V'', ''E'') be a graph with ''n'' = , ''V'', nodes and ''m'' = , ''E'', edges. For every subset ''U'' of vertices, its ''incidence vector'' 1''U'' is a vector of size ''n'', in which element ''v'' is 1 if node v is in ''U'', and 0 otherwise. Similarly, for every subset ''F'' of edges, its incidence vector 1F is a vector of size ''m'', in which element ''e'' is 1 if edge ''e'' is in ''F,'' and 0 otherwise. For every node ''v'' in ''V'', the set of edges in ''E'' adjacent to ''v'' is denoted by ''E''(''v''). Therefore, each vector 1''E(v)'' is a 1-by-''m'' vector in which element ''e'' is 1 if edge ''e'' is adjacent to ''v,'' and 0 otherwise. The ''incide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
