|
Wolfe Dual Problem
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] |
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] |
Philip Wolfe (mathematician)
Philip Starr "Phil" Wolfe (August 11, 1927 – December 29, 2016) was an American mathematician and one of the founders of convex optimization theory and mathematical programming. Life Wolfe received his bachelor's degree, masters, and Ph.D. degrees from the University of California, Berkeley. He and his wife, Hallie, lived in Ossining, New York. Career In 1954, he was offered an instructorship at Princeton, where he worked on generalizations of linear programming, such as quadratic programming and general non-linear programming, leading to the Frank–Wolfe algorithm in joint work with Marguerite Frank, then a visitor at Princeton. When Maurice Sion was on sabbatical at the Institute for Advanced Study, Sion and Wolfe published in 1957 an example of a zero-sum game without a minimax value. Wolfe joined RAND corporation in 1957, where he worked with George Dantzig, resulting in the now well known Dantzig–Wolfe decomposition method. In 1965, he moved to IBM's Thomas J ... [...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] |
Objective Function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of the func ... [...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] |
Lagrangian Duality
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] |
Infimum And 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 maxim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
KKT Conditions
KKT may refer to: * Karush–Kuhn–Tucker conditions, in mathematical optimization of nonlinear programming * kkt ( hu, közkereseti társaság, links=no), a type of general partnership in Hungary * Koi language, of Nepal, by ISO 639-3 code * Kappa Kappa Tau ''Scream Queens'' is an American satirical black comedy slasher television series that aired on Fox from September 22, 2015, to December 20, 2016. The series was created by Ryan Murphy, Brad Falchuk, and Ian Brennan and produced by Murphy, F ..., a fictional sorority in the television series '' Scream Queens'' {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fenchel Duality
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] |
