Supporting functional

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In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.

Convex analysis branch of mathematics devoted to the study of properties of convex functions and convex sets

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.

Mathematical optimization field in applied mathematics; the selection of a best element (with regard to some criterion) from some set of available alternatives

In mathematics, computer science and operations research, mathematical optimization or mathematical programming is the selection of a best element from some set of available alternatives.

Supporting hyperplane

In geometry, a supporting hyperplane of a set in Euclidean space is a hyperplane that has both of the following two properties:

Contents

Mathematical definition

Let X be a locally convex topological space, and be a convex set, then the continuous linear functional is a supporting functional of C at the point if and for every . [1]

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

Convex set (in convex geometry) subset of an affine space that is closed under convex combinations

In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations. More specifically, in a Euclidean space, a convex region is a region where, for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the region. 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.

Relation to support function

If (where is the dual space of ) is a support function of the set C, then if , it follows that defines a supporting functional of C at the point such that for any .

In mathematics, any vector space V has a corresponding dual vector space consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

In mathematics, the support functionhA of a non-empty closed convex set A in describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function on . Any non-empty closed convex set A is uniquely determined by hA. 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.

Relation to supporting hyperplane

If is a supporting functional of the convex set C at the point such that

then defines a supporting hyperplane to C at . [2]

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Reproducing kernel Hilbert space in functional analysis, a Hilbert space

In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions and in the RKHS are close in norm, i.e., is small, then and are also pointwise close, i.e., is small for all . The reverse need not be true.

In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation or Fenchel transformation. It is used to transform an optimization problem into its corresponding dual problem, which can often be simpler to solve.

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In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However in general the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.

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Convex cone

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In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure. The integral is also called the weak integral in contrast to the Bochner integral, which is the strong integral.

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References

  1. Pallaschke, Diethard; Rolewicz, Stefan (1997). Foundations of mathematical optimization: convex analysis without linearity. Springer. p. 323. ISBN   978-0-7923-4424-7.
  2. Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. p. 240. ISBN   978-0-387-29570-1.