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Exposed Point
In mathematics, an exposed point of a convex set C is a point x\in C at which some continuous linear functional attains its strict maximum over C. Such a functional is then said to ''expose'' x. There can be many exposing functionals for x. The set of exposed points of C is usually denoted \exp(C). A stronger notion is that of ''strongly exposed point'' of C which is an exposed point x \in C such that some exposing functional f of x attains its strong maximum over C at x, i.e. for each sequence (x_n) \subset C we have the following implication: f(x_n) \to \max f(C) \Longrightarrow \, x_n -x\, \to 0. The set of all strongly exposed points of C is usually denoted \operatorname\exp(C). There are two weaker notions, that of 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 ... [...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] |
