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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] |
Proper Convex Function
In mathematical analysis, in particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain, that never takes on the value -\infty and also is not identically equal to +\infty. In convex analysis and variational analysis, a point (in the domain) at which some given function f is minimized is typically sought, where f is valued in the extended real number line \infty, \infty= \mathbb \cup \. Such a point, if it exists, is called a of the function and its value at this point is called the () of the function. If the function takes -\infty as a value then -\infty is necessarily the global minimum value and the minimization problem can be answered; this is ultimately the reason why the definition of "" requires that the function never take -\infty as a value. Assuming this, if the function's domain is empty or if the function is identically equal to +\infty then the minimization problem once a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separated Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Pair
In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non-degenerate bilinear map b : X \times Y \to \mathbb. Duality theory, the study of dual systems, is part of functional analysis. According to Helmut H. Schaefer, "the study of a locally convex space in terms of its dual is the central part of the modern theory of topological vector spaces, for it provides the deepest and most beautiful results of the subject." Definition, notation, and conventions ;Pairings A or pair over a field \mathbb is a triple (X, Y, b), which may also be denoted by b(X, Y), consisting of two vector spaces X and Y over \mathbb (which this article assumes is either the real numbers or the complex numbers \Complex) and a bilinear map b : X \times Y \to \mathbb, which is called the bilinear map associated with the pairing or simply the pairing's map/bilinear form. For every x \in X, define \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (optimization)
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. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality. 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. This fact is called strong duality. Dual problem Usually the term "dual problem" refers to the ''Lagrangian dual problem'' but other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fenchel–Moreau Theorem
In convex analysis, the Fenchel–Moreau theorem (named after Werner Fenchel and Jean Jacques Moreau) or Fenchel biconjugation theorem (or just biconjugation theorem) is a theorem which gives necessary and sufficient conditions for a function to be equal to its biconjugate. This is in contrast to the general property that for any function f^ \leq f. This can be seen as a generalization of the bipolar theorem. It is used in duality theory to prove strong duality (via the perturbation function). Statement Let (X,\tau) be a Hausdorff locally convex space, for any extended real valued function f: X \to \mathbb \cup \ it follows that f = f^ if and only if one of the following is true # f is a proper, lower semi-continuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, ro ..., and c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lower Semi-continuous
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, roughly speaking, the function values for arguments near x_0 are not much higher (respectively, lower) than f\left(x_0\right). A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point x_0 to f\left(x_0\right) + c for some c>0, then the result is upper semicontinuous; if we decrease its value to f\left(x_0\right) - c then the result is lower semicontinuous. The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899. Definitions Assume throughout that X is a topological space and f:X\to\overline is a function with values in the extended real numbers \overline=\R \cup \ = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Function
In mathematical optimization, the perturbation function is any function which relates to primal and dual problems. The name comes from the fact that any such function defines a perturbation of the initial problem. In many cases this takes the form of shifting the constraints. In some texts the value function is called the perturbation function, and the perturbation function is called the bifunction. Definition Given two dual pairs of separated locally convex spaces \left(X,X^*\right) and \left(Y,Y^*\right). Then given the function f: X \to \mathbb \cup \, we can define the primal problem by :\inf_ f(x). \, If there are constraint conditions, these can be built into the function f by letting f \leftarrow f + I_\mathrm where I is the characteristic function. Then F: X \times Y \to \mathbb \cup \ is a ''perturbation function'' if and only if F(x,0) = f(x). Use in duality The duality gap is the difference of the right and left hand side of the inequality :\sup_ -F^*(0,y^*) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Duality
In applied mathematics, weak duality is a concept in optimization which states that the duality gap is always greater than or equal to 0. That means the solution to the dual (minimization) problem is ''always'' greater than or equal to the solution to an associated primal problem. This is opposed to strong duality which only holds in certain cases. Uses Many primal-dual approximation algorithms are based on the principle of weak duality.. Weak duality theorem The ''primal'' problem: : Maximize subject to ; The ''dual'' problem, : Minimize subject to . The weak duality theorem states . Namely, if (x_1,x_2,....,x_n) is a feasible solution for the primal maximization linear program and (y_1,y_2,....,y_m) is a feasible solution for the dual minimization linear program, then the weak duality theorem can be stated as \sum_^n c_j x_j \leq \sum_^m b_i y_i , where c_j and b_i are the coefficients of the respective objective functions. Proof: Generalizations More generally, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Duality
Strong duality is a condition in mathematical optimization in which the primal optimal objective and the dual optimal objective are equal. This is as opposed to weak duality (the primal problem has optimal value smaller than or equal to the dual problem, in other words the duality gap is greater than or equal to zero). Characterizations Strong duality holds if and only if the duality gap is equal to 0. Sufficient conditions Sufficient conditions comprise: * F = F^ where F is the perturbation function relating the primal and dual problems and F^ is the biconjugate of F (follows by construction of the duality gap) * F is convex and lower semi-continuous (equivalent to the first point by the Fenchel–Moreau theorem) * the primal problem is a linear optimization problem * Slater's condition for a convex optimization problem See also *Convex optimization Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Norm
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space. Definition Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous dual space. The dual norm of a continuous linear functional f belonging to X^* is the non-negative real number defined by any of the following equivalent formulas: \begin \, f \, &= \sup &&\ \\ &= \sup &&\ \\ &= \inf &&\ \\ &= \sup &&\ \\ &= \sup &&\ \;\;\;\text X \neq \ \\ &= \sup &&\bigg\ \;\;\;\text X \neq \ \\ \end where \sup and \inf denote the supremum and infimum, respectively. The constant 0 map is the origin of the vector space X^* and it always has norm \, 0\, = 0. If X = \ then the only linear functional on X is the constant 0 map and moreover, the sets in the last two rows will both be empty and consequently, their supremums will equal \sup \varnothing = - \infty instead of the correct value of 0. The ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
