Kantorovich inequality

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 p_i geq 0, 0 for i=1, dots ,n. :Let A_n={1,2,dots ,n}. :Then ::: left ( sum_{i=1}^n p_ix_i ight ) left (sum_{i=1}^n frac{p_i}{x_i} ight )

::leq frac{(a+b)^2}{4ab} left (sum_{i=1}^n p_i ight )^2-frac{(a-b)^2}{4ab} cdot min left{ left (sum_{i in X}p_i-sum_{j in Y}p_j ight )^2,:, {X cup Y=A_n},{X cap Y=varnothing} ight}.

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 Bunyakovsky inequality, the Wielandt inequality, and the Cauchy-Schwarz 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.


* [http://mathworld.wolfram.com/KantorovichInequality.html Eric W. Weisstein. "Kantorovich Inequality." From MathWorld--A Wolfram Web Resource.]

* [http://planetmath.org/encyclopedia/KantorovichInequality.html Planet Math entry on "Kantorovich inequality"]

* [http://carbon.cudenver.edu/~hgreenbe/glossary/index.php?page=K.html Mathematical Programming Glossary entry on "Kantorovich inequality"]

External links

* [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kantorovich.html Biography of Leonid Vitalyevich Kantorovich]

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Kantorovich theorem — The Kantorovich theorem is a mathematical statement on the convergence of Newton s method. It was first stated by Leonid Kantorovich in 1940. Newton s method constructs a sequence of points that with good luck will converge to a solution x of an… …   Wikipedia

  • List of Russian people — The Millennium of Russia monument in Veliky Novgorod, featuring the statues and reliefs of the most celebrated people in the first 1000 years of Russian history …   Wikipedia

  • List of Russian mathematicians — Andrey Kolmogorov, a preeminent 20th century mathematician. This list of Russian mathematicians includes the famous mathematicians from the Russian Empire, the Soviet Union and the Russian Federation. This list is incomplete; you can help by …   Wikipedia

  • List of mathematics articles (K) — NOTOC K K approximation of k hitting set K ary tree K core K edge connected graph K equivalence K factor error K finite K function K homology K means algorithm K medoids K minimum spanning tree K Poincaré algebra K Poincaré group K set (geometry) …   Wikipedia

  • List of inequalities — This page lists Wikipedia articles about named mathematical inequalities. Inequalities in pure mathematics =Analysis= * Askey–Gasper inequality * Bernoulli s inequality * Bernstein s inequality (mathematical analysis) * Bessel s inequality *… …   Wikipedia

  • Неравенство Швейцера — гласит следующее Для любых вещественных чисел , принадлежащих отрезку , где , имеет место неравенство Более того, если …   Википедия

  • Mathematical economics — Economics …   Wikipedia

  • Linear programming — (LP, or linear optimization) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships.… …   Wikipedia

  • John von Neumann — Von Neumann redirects here. For other uses, see Von Neumann (disambiguation). The native form of this personal name is Neumann János. This article uses the Western name order. John von Neumann …   Wikipedia

  • optimization — /op teuh meuh zay sheuhn/ 1. the fact of optimizing; making the best of anything. 2. the condition of being optimized. 3. Math. a mathematical technique for finding a maximum or minimum value of a function of several variables subject to a set of …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.