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Fenchel's Duality Theorem
In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel. Let ''ƒ'' be a proper convex function on R''n'' and let ''g'' be a proper concave function on R''n''. Then, if regularity conditions are satisfied, :\inf_x ( f(x)-g(x) ) = \sup_p ( g_*(p)-f^*(p)). where ''ƒ'' * is the convex conjugate of ''ƒ'' (also referred to as the Fenchel–Legendre transform) and ''g'' * is the concave conjugate of ''g''. That is, :f^ \left( x^ \right) := \sup \left \ :g_ \left( x^ \right) := \inf \left \ Mathematical theorem Let ''X'' and ''Y'' be Banach spaces, f: X \to \mathbb \cup \ and g: Y \to \mathbb \cup \ be convex functions and A: X \to Y be a bounded linear map. Then the Fenchel problems: :p^* = \inf_ \ :d^* = \sup_ \ satisfy weak duality, i.e. p^* \geq d^*. Note that f^*,g^* are the convex conjugates of ''f'',''g'' respectively, and A^* is the adjoint operator. The perturbation function for this du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Werner Fenchel
Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the foundation for nonlinear programming. A German-born Jew and early refugee from Nazi suppression of intellectuals, Fenchel lived most of his life in Denmark. Fenchel's monographs and lecture notes are considered influential. Biography Early life and education Fenchel was born on 3 May 1905 in Berlin, Germany, his younger brother was the Israeli film director and architect Heinz Fenchel. Fenchel studied mathematics and physics at the University of Berlin between 1923 and 1928. He wrote his doctorate thesis in geometry (''Über Krümmung und Windung geschlossener Raumkurven'') under Ludwig Bieberbach. Professorship in Germany From 1928 to 1933, Fenchel was Professor E. Landau's Assistant at the Univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Interior
In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior. Definition Assume that A is a subset of a vector space X. The ''algebraic interior'' (or ''radial kernel'') ''of A with respect to X'' is the set of all points at which A is a radial set. A point a_0 \in A is called an of A and A is said to be if for every x \in X there exists a real number t_x > 0 such that for every t \in , t_x a_0 + t x \in A. This last condition can also be written as a_0 + , t_xx \subseteq A where the set a_0 + , t_xx ~:=~ \left\ is the line segment (or closed interval) starting at a_0 and ending at a_0 + t_x x; this line segment is a subset of a_0 + radial points of the set. If M is a linear subspace of X and A \subseteq X then this definition can be generalized to the ''algebraic interior of A with respect to M'' is: \operatorname_M A := \left\. where \operatorname_M A \subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfe Duality
In mathematical optimization, Wolfe duality, named after Philip Wolfe, is type of dual problem in which the objective function and constraints are all differentiable functions. Using this concept a lower bound for a minimization problem can be found because of the weak duality principle. Mathematical formulation For a minimization problem with inequality constraints, : \begin &\underset& & f(x) \\ &\operatorname & &g_i(x) \leq 0, \quad i = 1,\dots,m \end the Lagrangian dual problem is : \begin &\underset& & \inf_x \left(f(x) + \sum_^m u_j g_j(x)\right) \\ &\operatorname & &u_i \geq 0, \quad i = 1,\dots,m \end where the objective function is the Lagrange dual function. Provided that the functions f and g_1, \ldots, g_m are convex and continuously differentiable, the infimum occurs where the gradient is equal to zero. The problem : \begin &\underset& & f(x) + \sum_^m u_j g_j(x) \\ &\operatorname & & \nabla f(x) + \sum_^m u_j \nabla g_j(x) = 0 \\ &&&u_i \geq 0, \quad i = 1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moreau's Theorem
In mathematics, Moreau's theorem is a result in convex analysis named after French mathematician Jean-Jacques Moreau. It shows that sufficiently well-behaved convex functionals on Hilbert spaces are differentiable and the derivative is well-approximated by the so-called Yosida approximation, which is defined in terms of the resolvent operator. Statement of the theorem Let ''H'' be a Hilbert space and let ''φ'' : ''H'' → R ∪ be a proper function, proper, convex and semi-continuity, lower semi-continuous extended real number line, extended real-valued functional on ''H''. Let ''A'' stand for ∂''φ'', the subderivative of ''φ''; for ''α'' > 0 let ''J''''α'' denote the resolvent: :J_ = (\mathrm + \alpha A)^; and let ''A''''α'' denote the Yosida approximation to ''A'': :A_ = \frac1 ( \mathrm - J_ ). For each ''α'' > 0 and ''x'' ∈ ''H'', let :\varphi_ (x) = \inf_ \f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Conjugate
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, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality. Definition Let X be a real topological vector space and let X^ be the dual space to X. Denote by :\langle \cdot , \cdot \rangle : X^ \times X \to \mathbb the canonical dual pairing, which is defined by \left( x^*, x \right) \mapsto x^* (x). For a function f : X \to \mathbb \cup \ taking values on the extended real number line, its is the function :f^ : X^ \to \mathbb \cup \ whose value at x^* \in X^ is defined to be the supremum: :f^ \left( x^ \right) := \sup \left\, or, equivalently, in terms of the infimum: :f^ \left( x^ \right) := - \inf \left\. This definition can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of one quantity (such as velocity, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism (or vice versa) and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables. For sufficiently smooth functions on the real line, the Legendre transform f^* of a function f can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other. This can be expressed in Euler's derivative notation as Df(\cdot) = \left( D f^* \right) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and max ... [...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] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...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] |
