“Kantorovich inequality”的意思、由来-开放百科全书

词条

Kantorovich inequality

释义

References
External links

In mathematics, the Kantorovich inequality is a particular case of the Cauchy–Schwarz inequality, which is itself a generalization of the triangle inequality.

The triangle inequality states that the length of two sides of any triangle, added together, will be equal to or greater than the length of the third side. In simplest terms, the Kantorovich inequality translates the basic idea of the triangle inequality into the terms and notational conventions of linear programming. (See vector space, inner product, and normed vector space for other examples of how the basic ideas inherent in the triangle inequality—line segment and distance—can be generalized into a broader context.)

More formally, the Kantorovich inequality can be expressed this way:

Let

Then

The Kantorovich inequality is used in convergence analysis; it bounds the convergence rate of Cauchy's steepest descent.

Equivalents of the Kantorovich inequality have arisen in a number of different fields. For instance, the Cauchy–Schwarz–Bunyakovsky inequality and the Wielandt inequality are equivalent to the Kantorovich inequality and all of these are, in turn, special cases of the Hölder inequality.

The Kantorovich inequality is named after Soviet economist, mathematician, and Nobel Prize winner Leonid Kantorovich, a pioneer in the field of linear programming.

There is also Matrix version of the Kantrovich inequality due to Marshall and Olkin.

References

{{MathWorld|urlname=KantorovichInequality|title=Kantorovich Inequality}}
{{PlanetMath|urlname=KantorovichInequality|title=Cauchy-Schwarz inequality}}
Mathematical Programming Glossary entry on "Kantorovich inequality"
MARSHALL A. W. and OLKIN, I., Matrix versions of the Cauchy and Kantorovieh inequatities. Aequationes Mathematicae 40 (1990), pp. 89–93.

External links

Biography of Leonid Vitalyevich Kantorovich

2 : Theorems in analysis|Inequalities

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。