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Semi-infinite Programming
In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraints. In the former case the constraints are typically parameterized. * * M. A. Goberna and M. A. López, ''Linear Semi-Infinite Optimization'', Wiley, 1998. * Mathematical formulation of the problem The problem can be stated simply as: : \min_\;\; f(x) : \text :: g(x,y) \le 0, \;\; \forall y \in Y where :f: R^n \to R :g: R^n \times R^m \to R :X \subseteq R^n :Y \subseteq R^m. SIP can be seen as a special case of bilevel programs in which the lower-level variables do not participate in the objective function. Methods for solving the problem In the meantime, see external links below for a complete tutorial. Examples In the meantime, see external links below for a complete tutorial. See also * Optimization Mathematical optimization (alternativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization (mathematics)
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] |
Optimization Problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: * An optimization problem with discrete variables is known as a '' discrete optimization'', in which an object such as an integer, permutation or graph must be found from a countable set. * A problem with continuous variables is known as a ''continuous optimization'', in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems. Continuous optimization problem The '' standard form'' of a continuous optimization problem is \begin &\underset& & f(x) \\ &\operatorname & &g_i(x) \leq 0, \quad i = 1,\dots,m \\ &&&h_j(x) = 0, \quad j = 1, \dots,p \end where * is the objective function to be minimized over the -variable vector , * are called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilevel Program
Bilevel optimization is a special kind of optimization where one problem is embedded (nested) within another. The outer optimization task is commonly referred to as the upper-level optimization task, and the inner optimization task is commonly referred to as the lower-level optimization task. These problems involve two kinds of variables, referred to as the upper-level variables and the lower-level variables. Mathematical formulation of the problem A general formulation of the bilevel optimization problem can be written as follows: \min\limits_\;\; F(x,y) subject to: G_i(x,y) \leq 0, for i \in \ y \in \arg \min \limits_ \ where : F,f: R^ \times R^ \to R : G_i,g_j: R^ \times R^ \to R : X \subseteq R^ : Y \subseteq R^. In the above formulation, F represents the upper-level objective function and f represents the lower-level objective function. Similarly x represents the upper-level decision vector and y represents the lower-level decision vector. G_i and g_j represent th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Semi-infinite Programming
In mathematics, a semi-infinite programming (SIP) problem is an optimization problem with a finite number of variables and an infinite number of constraints. The constraints are typically parameterized. In a generalized semi-infinite programming (GSIP) problem, the feasible set of the parameters depends on the variables.O. Stein and G. Still, On generalized semi-infinite optimization and bilevel optimization', European J. Oper. Res., 142 (2002), pp. 444-462 Mathematical formulation of the problem The problem can be stated simply as: : \min\limits_\;\; f(x) : \mbox\ :: g(x,y) \le 0, \;\; \forall y \in Y(x) where :f: R^n \to R :g: R^n \times R^m \to R :X \subseteq R^n :Y \subseteq R^m. In the special case that the set :Y(x) is nonempty for all x \in X GSIP can be cast as bilevel programs ( Multilevel programming). Methods for solving the problem Examples See also * optimization * Semi-Infinite Programming (SIP) References {{Reflist External linksMathematica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization In Vector Spaces
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying comput ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
