Criss-cross Algorithm

picture info	Criss-cross Algorithm In mathematical optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear inequality constraints and nonlinear objective functions; there are criss-cross algorithms for linear-fractional programming problems, quadratic-programming problems, and linear complementarity problems. Like the simplex algorithm of George B. Dantzig, the criss-cross algorithm is not a polynomial-time algorithm for linear programming. Both algorithms visit all 2''D'' corners of a (perturbed) cube in dimension ''D'', the Klee–Minty cube (after Victor Klee and George J. Minty), in the worst case. However, when it is started at a random corner, the criss-cross algorithm on average visits only ''D'' additional corners.The simplex algorithm takes on average ''D'' steps for a cube. : Thus, for the three-dimensional cube, the algorithm visits all 8 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Unitcube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.. See in particulap. 671. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.. Unit hypercube The term ''unit cube'' or unit hypercube is also used for hypercubes, or "cubes" in ''n''-dimensional spaces, for values of ''n'' other than 3 and edge length 1. Sometimes the term "unit cube" refers in specific to the set , 1sup>''n'' of all ''n''-tuples of numbers in the interval , 1 The length of the longest diagonal of a unit hypercube of ''n'' dimensions is \sqrt n, the square root of ''n'' and the (Euclidean) length of the vector (1,1,1,....1,1) in ''n''-dimensional space. See also Doubling the cube K-cell Robbins constant, the average distance between two random points in a unit cube Tychonoff cube, an infinite-dimensional analogue of the unit cube Unit square Unit sphere In mathematics, a unit sphere is simply a sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Tamas Terlaky Tamas may refer to: * ''Tamas'' (philosophy), a concept of darkness and death in Hindu philosophy * Tamás (name), a given name in Hungarian (Thomas) * ''Tamas'' (film), a 1987 TV series/movie directed by Govind Nihalani * ''Tamas'' (novel), a 1975 novel by Bhisham Sahni * Christian Tămaș, Romanian writer * Gabriel Tamaș Gabriel Sebastian Tamaș (; born 9 November 1983) is a Romanian professional footballer who plays as a centre-back for Liga II club Concordia Chiajna. Tamaș started out as a senior at FC Brașov in the 1998–99 season, and is a journeyman w ... (born 1983), Romanian footballer * Vladimir Tămaș, Romanian footballer See also * Tama (other) {{Disambiguation, surname hu:Tamás ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Ellipsoid 2 An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is bounded, which means that it may be enclosed in a sufficiently large sphere. An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the ''principal axes'', or simply axes of the ellipsoid. If the three axes have different lengths, the figure is a triaxial ellipsoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Extreme Point In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or corner point of S. Definition Throughout, it is assumed that X is a real or complex vector space. For any p, x, y \in X, say that p x and y if x \neq y and there exists a 0 < t < 1 such that $p = t x + (1-t) y.$ If $K$ is a subset of $X$ and $p \in K,$ then $p$ is called an of $K$ if it does not lie between any two points of $K.$ That is, if there does exist $x, y \in K$ and $0 < t < 1$ such that $x \neq y$ and $p = t x + (1-t) y.$ The set of all extreme points of $K$ is denoted by $\operatorname(K).$ Generalizations If $S$ is a subset of a vector space then a linear sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Krein–Milman Theorem In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape. This observation also holds for any other convex polygon in the plane \R^2. Statement and definitions Preliminaries and definitions Throughout, X will be a real or complex vector space. For any elements x and y in a vector space, the set , y:= \ is called the or closed interval between x and y. The or open interval between x and y is (x, x) := \varnothing when x = y while it is (x, y) := \ when x \neq y; it satisfies (x, y) = , y\setminus \ and , y= (x, y) \cup \. The points x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Halfspace Half-space may refer to: * Half-space (geometry), either of the two parts into which a plane divides Euclidean space * Half-space (punctuation), a spacing character half the width of a regular space * (Poincaré) Half-space model, a model of 3-dimensional hyperbolic geometry or higher-dimensional hyperbolic geometry using a Euclidean half-space * Half-space model (oceanography), an estimate for seabed height in areas without significant subduction * Siegel upper half-space In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It ... , a set of complex matrices with positive definite imaginary part {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Weyl's Theorem In mathematics, Weyl's theorem or Weyl's lemma might refer to one of a number of results of Hermann Weyl. These include * the Peter–Weyl theorem * Weyl's theorem on complete reducibility, results originally derived from the unitarian trick on representation theory of semisimple groups and semisimple Lie algebras * Weyl's theorem on eigenvalues * Weyl's criterion for equidistribution (Weyl's criterion) * Weyl's lemma on the hypoellipticity of the Laplace equation * results estimating Weyl sums in the theory of exponential sums * Weyl's inequality In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix. Weyl's inequality about perturbation Let ... * Weyl's criterion for a number to be in the essential spectrum of an operator {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Farkas Lemma Farkas' lemma is a solvability theorem for a finite system of linear inequalities in mathematics. It was originally proven by the Hungarian mathematician Gyula Farkas. Farkas' lemma is the key result underpinning the linear programming duality and has played a central role in the development of mathematical optimization (alternatively, mathematical programming). It is used amongst other things in the proof of the Karush–Kuhn–Tucker theorem in nonlinear programming. Remarkably, in the area of the foundations of quantum theory, the lemma also underlies the complete set of Bell inequalities in the form of necessary and sufficient conditions for the existence of a local hidden-variable theory, given data from any specific set of measurements. Generalizations of the Farkas' lemma are about the solvability theorem for convex inequalities, i.e., infinite system of linear inequalities. Farkas' lemma belongs to a class of statements called "theorems of the alternative": a theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Steinitz's Theorem In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. This result provides a classification theorem for the three-dimensional convex polyhedra, something that is not known in higher dimensions. It provides a complete and purely combinatorial description of the graphs of these polyhedra, allowing other results on them, such as Eberhard's theorem on the realization of polyhedra with given types of faces, to be proven more easily, without reference to the geometry of these shapes. Additionally, it has been applied in graph drawing, as a way to construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Linear Algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Real Number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]