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Supporting Functional
In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set. Mathematical definition Let ''X'' be a locally convex topological space, and C \subset X be a convex set, then the continuous linear functional \phi: X \to \mathbb is a supporting functional of ''C'' at the point x_0 if \phi \not=0 and \phi(x) \leq \phi(x_0) for every x \in C. Relation to support function If h_C: X^* \to \mathbb (where X^* is the dual space of X) is a support function In mathematics, the support function ''h'A'' of a non-empty closed convex set ''A'' in \mathbb^n describes the (signed) distances of supporting hyperplanes of ''A'' from the origin. The support function is a convex function on \mathbb^n. Any n ... of the set ''C'', then if h_C\left(x^*\right) = x^*\left(x_0\right), it follows that h_C defines a supporting functional \phi: X \to \mathbb of ''C'' at the point x_0 such that \phi(x) = x^*(x) for any x \in X. Relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of some vector space X is if it satisfies any of the following equivalent conditions: #If 0 \leq r \leq 1 is real and x, y \in C then r x + (1 - r) y \in C. #If 0 is a if holds for any real 0 is called if \operatorname f \neq \varnothing and f(x) > -\infty for x \in \operatorname f. Alternatively, this means that there exists some x in the domain of f at which f(x) \in \mathbb and f is also equal to -\infty. In words, a function is if its domain is not empty, it never takes on the value -\infty, and it also is not identically equal to +\infty. If f : \mathbb^n \to \infty, \infty/math> is a proper convex function then there exist some vector b \in \mathbb^n and some r \in \mathbb such that :f(x) \geq x \cdot b - r for every x where ... [...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] |
Supporting Hyperplane
In geometry, a supporting hyperplane of a set S in Euclidean space \mathbb R^n is a hyperplane that has both of the following two properties: * S is entirely contained in one of the two closed half-spaces bounded by the hyperplane, * S has at least one boundary-point on the hyperplane. Here, a closed half-space is the half-space that includes the points within the hyperplane. Supporting hyperplane theorem This theorem states that if S is a convex set in the topological vector space X=\mathbb^n, and x_0 is a point on the boundary of S, then there exists a supporting hyperplane containing x_0. If x^* \in X^* \backslash \ (X^* is the dual space of X, x^* is a nonzero linear functional) such that x^*\left(x_0\right) \geq x^*(x) for all x \in S, then :H = \ defines a supporting hyperplane. Conversely, if S is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then S is a convex set. The hyperplane in the theorem may not be uniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. History Metrizable topologies on vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Linear Functional
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous linear operators Characterizations of continuity Suppose that F : X \to Y is a linear operator between two topological vector spaces (TVSs). The following are equivalent: F is continuous. F is continuous at some point x \in X. F is continuous at the origin in X. if Y is locally convex then this list may be extended to include: for every continuous seminorm q on Y, there exists a continuous seminorm p on X such that q \circ F \leq p. if X and Y are both Hausdorff locally convex spaces then this list may be extended to include: F is weakly continuous and its transpose ^t F : Y^ \to X^ maps equicontinuous subsets of Y^ to equicontinuous subsets of X^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support Function
In mathematics, the support function ''h''''A'' of a non-empty closed convex set ''A'' in \mathbb^n describes the (signed) distances of supporting hyperplanes of ''A'' from the origin. The support function is a convex function on \mathbb^n. Any non-empty closed convex set ''A'' is uniquely determined by ''h''''A''. Furthermore, the support function, as a function of the set ''A'', is compatible with many natural geometric operations, like scaling, translation, rotation and Minkowski addition. Due to these properties, the support function is one of the most central basic concepts in convex geometry. Definition The support function h_A\colon\mathbb^n\to\mathbb of a non-empty closed convex set ''A'' in \mathbb^n is given by : h_A(x)=\sup\, x\in\mathbb^n; see T. Bonnesen, W. Fenchel, '' Theorie der konvexen Körper,'' Julius Springer, Berlin, 1934. English translation: ''Theory of convex bodies,'' BCS Associates, Moscow, ID, 1987. R. J. Gardner, ''Geometric tomography,'' Cam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality Theories
Duality may refer to: Mathematics * Duality (mathematics), a mathematical concept ** Dual (category theory), a formalization of mathematical duality ** Duality (optimization) ** Duality (order theory), a concept regarding binary relations ** Duality (projective geometry), general principle of projective geometry ** Duality principle (Boolean algebra), the extension of order-theoretic duality to Boolean algebras ** S-duality (homotopy theory) * Philosophy, logic, and psychology * Dualistic cosmology, a twofold division in several spiritual and religious worldviews * Dualism (philosophy of mind), where the body and mind are considered to be irreducibly distinct * De Morgan's laws, specifically the ability to generate the dual of any logical expression * Complementary duality of Carl Jung's functions and types in Socionics Science Electrical and mechanical * Duality (electrical circuits), regarding isomorphism of electrical circuits * Duality (mechanical engineering), regarding i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
