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Max–min Inequality
In mathematics, the max–min inequality is as follows: :For any function \ f : Z \times W \to \mathbb\ , :: \sup_ \inf_ f(z, w) \leq \inf_ \sup_ f(z, w)\ . When equality holds one says that , , and satisfies a strong max–min property (or a saddle-point property). The example function \ f(z,w) = \sin( z + w )\ illustrates that the equality does not hold for every function. A theorem giving conditions on , , and which guarantee the saddle point property is called a minimax theorem. Proof Define g(z) \triangleq \inf_ f(z, w)\ . For all z \in Z, we get g(z) \leq f(z, w) for all w \in W by definition of the infimum being a lower bound. Next, for all w \in W , f(z, w) \leq \sup_ f(z, w) for all z \in Z by definition of the supremum being an upper bound. Thus, for all z \in Z and w \in W , g(z) \leq f(z, w) \leq \sup_ f(z, w) making h(w) \triangleq \sup_ f(z, w) an upper bound on g(z) for any choice of w \in W . Because the supremum is the least upper bound, \sup_ g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle Point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f(x,y) = x^2 + y^3 has a critical point at (0, 0) that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y-direction. The name derives from the fact that the prototypical example in two dimensions is a surface that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimax Theorem
In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality. The first theorem in this sense is von Neumann's minimax theorem from 1928, which was considered the starting point of game theory. Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature. Zero-sum games The minimax theorem was first proven and published in 1928 by John von Neumann, who is quoted as saying "''As far as I can see, there could be no theory of games ... without that theorem ... I thought there was nothing worth publishing until the Minimax Theorem was proved''". Formally, von Neumann's minimax theorem states: Let X \subset \mathbb^n and Y \subset \mathbb^m be compact convex sets. If f: X \times Y \rightarrow \mathbb is a continuous function that is concave-convex, i.e. : f(\cdot,y): X \to \mathbb is concave for fixed y, and : f(x,\cdo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimax Theorem
In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality. The first theorem in this sense is von Neumann's minimax theorem from 1928, which was considered the starting point of game theory. Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature. Zero-sum games The minimax theorem was first proven and published in 1928 by John von Neumann, who is quoted as saying "''As far as I can see, there could be no theory of games ... without that theorem ... I thought there was nothing worth publishing until the Minimax Theorem was proved''". Formally, von Neumann's minimax theorem states: Let X \subset \mathbb^n and Y \subset \mathbb^m be compact convex sets. If f: X \times Y \rightarrow \mathbb is a continuous function that is concave-convex, i.e. : f(\cdot,y): X \to \mathbb is concave for fixed y, and : f(x,\cdo ... [...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 maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
