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Maximum Theorem
The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959. The theorem is primarily used in mathematical economics and optimal control. Statement of theorem Maximum Theorem. Let X and \Theta be topological spaces, f:X\times\Theta\to\mathbb be a continuous function on the product X \times \Theta, and C:\Theta\rightrightarrows X be a compact-valued correspondence such that C(\theta) \ne \emptyset for all \theta \in \Theta. Define the ''marginal function'' (or ''value function'') f^* : \Theta \to \mathbb by :f^*(\theta)=\sup\ and the ''set of maximizers'' C^* : \Theta \rightrightarrows X by : C^*(\theta)= \mathrm\max\ = \ . If C is continuous (i.e. both upper and lower hemicontinuous) at \theta, then the value function f^* is continuous, and the set of maximizers C^* is upper-hemicontinuous with nonempty and compact values. As a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies 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 . 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 most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Utility Function
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a Normative economics, normative context, utility refers to a goal or objective that we wish to maximize, i.e., an objective function. This kind of utility bears a closer resemblance to the original Utilitarianism, utilitarian concept, developed by moral philosophers such as Jeremy Bentham and John Stuart Mill. * In a Positive economics, descriptive context, the term refers to an ''apparent'' objective function; such a function is Revealed preference, revealed by a person's behavior, and specifically by their preferences over Lottery (decision theory), lotteries, which can be any quantified choice. The relationship between these two kinds of utility functions has been a source of controversy among both Economics, economists and Ethics, ethicists, with most maintaining that the two are distinct but generally re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematical Optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems 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. Optimization problems Opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematical Economics
Mathematical economics is the application of Mathematics, mathematical methods to represent theories and analyze problems in economics. Often, these Applied mathematics#Economics, applied methods are beyond simple geometry, and may include differential and integral calculus, Recurrence relation, difference and differential equations, Matrix (mathematics), matrix algebra, mathematical programming, or other Computational economics, computational methods.TOC. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity. Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Convex Optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard. Definition Abstract form A convex optimization problem is defined by two ingredients: * The ''objective function'', which is a real-valued convex function of ''n'' variables, f :\mathcal D \subseteq \mathbb^n \to \mathbb; * The ''feasible set'', which is a convex subset C\subseteq \mathbb^n. The goal of the problem is to find some \mathbf \in C attaining :\inf \. In general, there are three options regarding the existence of a solution: * If such a point ''x''* exists, it is referred to as an ''optimal point'' or ''solution''; the set of all optimal points is called the ''optimal set''; and the problem is called ''solvable''. * If f is unbou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Theory Of Continuous Functions
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, and research. Theories can be scientific, falling within the realm of empirical and testable knowledge, or they may belong to non-scientific disciplines, such as philosophy, art, or sociology. In some cases, theories may exist independently of any formal discipline. In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction (" falsify") of it. Scientific theories are the most reliable, rigorous, and comprehensive form of scientif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Michael Selection Theorem
In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following: Conversely, if any lower semicontinuous multimap from topological space ''X'' to a Banach space, with nonempty convex closed values, admits a continuous selection, then ''X'' is paracompact. This provides another characterization for paracompactness. Examples A function that satisfies all requirements The function: F(x)= -x/2, ~1-x/4, shown by the grey area in the figure at the right, is a set-valued function from the real interval ,1to itself. It satisfies all Michael's conditions, and indeed it has a continuous selection, for example: f(x)= 1-x/2 or f(x)= 1-3x/8 . A function that does not satisfy lower hemicontinuity The function F(x)= \begin 3/4 & 0 \le x < 0.5 \\ \left ,1\right & x = 0.5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Envelope Theorem
In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in the objective function. The envelope theorem is an important tool for comparative statics of optimization models. The term envelope derives from describing the graph of the value function as the "upper envelope" of the graphs of the parameterized family of functions \left\ _ that are optimized. Statement Let f(x,\alpha) and g_(x,\alpha), j = 1,2, \ldots, m be real-valued continuously differentiable functions on \mathbb^, where x \in \mathbb^ are choice variables and \alpha \in \mathbb^ are parameters, and consider the problem of choosing x, for a given \alpha, so as to: : \max_ f(x, \alpha) subject to g_(x,\alpha) \geq 0, j = 1,2, \ldots, m and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Kakutani Fixed-point Theorem
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions. The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics. Statement Kakutani's theorem states: : ''Let'' ''S'' ''be a non-empty, compact and convex subset of some Euclidean space'' R''n''. :''Let'' ''φ'':&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Brouwer Fixed-point Theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a nonempty compactness, compact convex set to itself, there is a point x_0 such that f(x_0)=x_0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed Disk (mathematics), disk D to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset K of Euclidean space to itself. Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem. This gives it a place ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
General Equilibrium Theory
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall general equilibrium. General equilibrium theory contrasts with the theory of ''partial'' equilibrium, which analyzes a specific part of an economy while its other factors are held constant. General equilibrium theory both studies economies using the model of equilibrium pricing and seeks to determine in which circumstances the assumptions of general equilibrium will hold. The theory dates to the 1870s, particularly the work of French economist Léon Walras in his pioneering 1874 work ''Elements of Pure Economics''. The theory reached its modern form with the work of Lionel W. McKenzie (Walrasian theory), Kenneth Arrow and Gérard Debreu (Hicksian theory) in the 1950s. Overview Broadly speaking, general equilibrium tries to give ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
