In convex analysis, a branch of mathematics, the effective domain is an extension of the domain of a function defined for functions that take values in the extended real number line In convex analysis and variational analysis, a point at which some given extended real-valued function is minimized is typically sought, where such a point is called a global minimum point. The effective domain of this function is defined to be the set of all points in this function's domain at which its value is not equal to where the effective domain is defined this way because it is only these points that have even a remote chance of being a global minimum point. Indeed, it is common practice in these fields to set a function equal to at a point specifically to exluded that point from even being considered
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| - Effective domain (en)
- Domaine effectif (fr)
- 有効領域 (ja)
- 有效域 (zh)
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| - En mathématiques, et plus précisément en analyse convexe, le domaine effectif d'une fonction à valeurs dans la droite réelle achevée est l'ensemble des points où elle ne prend pas la valeur . (fr)
- 数学の一分野である凸解析において、有効領域（ゆうこうりょういき、英: effective domain）は、定義域の概念を拡張したものである。 ベクトル空間 X が与えられたとき、拡大実数を値域とする凸函数 は、次で定義される有効領域を持つ： この函数が凹函数である場合、有効領域は次のようになる： 有効領域は、函数 のエピグラフの X の上への射影と等しい。すなわち、次で与えられる。 凸函数が通常の実数への写像 であるなら、有効領域は通常の定義域と一致する。 函数 が真凸函数であるための必要十分条件は、f が凸で、f の有効領域が空でなく、すべての に対して が成立することである。 (ja)
- 在数学的一个分支——中，有效域是对定义域的扩展。 (zh)
- In convex analysis, a branch of mathematics, the effective domain is an extension of the domain of a function defined for functions that take values in the extended real number line In convex analysis and variational analysis, a point at which some given extended real-valued function is minimized is typically sought, where such a point is called a global minimum point. The effective domain of this function is defined to be the set of all points in this function's domain at which its value is not equal to where the effective domain is defined this way because it is only these points that have even a remote chance of being a global minimum point. Indeed, it is common practice in these fields to set a function equal to at a point specifically to exluded that point from even being considered (en)
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| - In convex analysis, a branch of mathematics, the effective domain is an extension of the domain of a function defined for functions that take values in the extended real number line In convex analysis and variational analysis, a point at which some given extended real-valued function is minimized is typically sought, where such a point is called a global minimum point. The effective domain of this function is defined to be the set of all points in this function's domain at which its value is not equal to where the effective domain is defined this way because it is only these points that have even a remote chance of being a global minimum point. Indeed, it is common practice in these fields to set a function equal to at a point specifically to exluded that point from even being considered as a potential solution (to the minimization problem). Points at which the function takes the value (if any) belong to the effective domain because such points are considered acceptable solutions to the minimization problem, with the reasoning being that if such a point was not acceptable as a solution then the function would have already been set to at that point instead. When a minimum point (in ) of a function is to be found but 's domain is a proper subset of some vector space then it often technically useful to extend to all of by setting at every By definition, no point of belongs to the effective domain of which is consistent with the desire to find a minimum point of the original function rather than of the newly defined extension to all of If the problem is instead a maximization problem (which would be clearly indicated) then the effective domain instead consists of all points in the function's domain at which it is not equal to (en)
- En mathématiques, et plus précisément en analyse convexe, le domaine effectif d'une fonction à valeurs dans la droite réelle achevée est l'ensemble des points où elle ne prend pas la valeur . (fr)
- 数学の一分野である凸解析において、有効領域（ゆうこうりょういき、英: effective domain）は、定義域の概念を拡張したものである。 ベクトル空間 X が与えられたとき、拡大実数を値域とする凸函数 は、次で定義される有効領域を持つ： この函数が凹函数である場合、有効領域は次のようになる： 有効領域は、函数 のエピグラフの X の上への射影と等しい。すなわち、次で与えられる。 凸函数が通常の実数への写像 であるなら、有効領域は通常の定義域と一致する。 函数 が真凸函数であるための必要十分条件は、f が凸で、f の有効領域が空でなく、すべての に対して が成立することである。 (ja)
- 在数学的一个分支——中，有效域是对定义域的扩展。 (zh)
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