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{{dablink|This article is related to [[mathematical programming]]. For other uses see [[complementarity (disambiguation)|complementarity]].}}
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A '''complementarity problem''' is a type of [[mathematical optimization]] problem. It is the problem of optimizing (minimizing or maximizing) a function of two [[vector space|vector]] variables subject to certain requirements (constraints) which include: that the [[inner product]] of the two vectors must equal zero, i.e. orthoginal.<ref>{{Cite journal | last1=Billups | first1=Stephen | last2=Murty | first2=Katta | title=Complementarity Problems | doi=10.1016/S0377-0427(00)00432-5|url=http://www-personal.umich.edu/~murty/LCPart.ps | year=2000 | journal=Journal of Computational and Applied Mathematics | volume=124 | pages=303}} </ref> In particular for finite-dimensional real vector spaces this means that, if one has vectors ''X'' and ''Y'' with ''nonnegative'' components (''x''<sub>''i''</sub>&nbsp;≥&nbsp;0 and ''y''<sub>''i''</sub>&nbsp;≥&nbsp;0 for all <math>i</math>: in the [[first quadrant]] if 2-dimensional, in the first [[octant (solid geometry)|octant]] if 3-dimensional), then for each pair of components ''x''<sub>''i''</sub> and ''y''<sub>''i''</sub> one of the pair must be zero, hence the name ''complementarity''. e.g. ''X''&nbsp;=&nbsp;(1,&nbsp;0) and ''Y''&nbsp;=&nbsp;(0,&nbsp;2) are complementary, but ''X''&nbsp;=&nbsp;(1,&nbsp;1) and ''Y''&nbsp;=&nbsp;(2,&nbsp;0) are not. A complementarity problem is a special case of a [[variational inequality]].
 
==History==
Complementarity problems were originally studied because the [[Karush–Kuhn–Tucker conditions]] in [[linear programming]] and [[quadratic programming]] constitute a [[linear complementarity problem]] (LCP) or a [[mixed complementarity problem]] (MCP). In 1963 [[Carlton E. Lemke|Lemke]] and [[J.T. Howson|Howson]] showed that, for two person games, computing a [[Nash equilibrium]] point is equivalent to an LCP. In 1968 [[Richard W. Cottle|Cottle]] and [[George B. Dantzig|Dantzig]] unified linear and quadratic programming and [[bimatrix game]]s. Since then the study of complementarity problems and variational inequalities has expanded enormously.
 
Areas of [[mathematics]] and [[science]] that contributed to the development of complementarity theory
include: [[Optimization (mathematics)|optimization]], [[equilibrium point|equilibrium]] problems, [[Variational inequality|variational inequality theory]], [[fixed point theory]], [[topological degree theory]] and [[nonlinear analysis]].
 
==See also==
* [[Mathematical programming with equilibrium constraints]]
* [[nl (format)|nl format]] for representing complementarity problems
 
==References==
<references/>
 
==Further reading==
 
* {{cite book | author=Richard W. Cottle, Jong-Shi Pang, Richard E. Stone | title=The Linear Complementarity Problem
| publisher=Academic Press | year=1992 | isbn=978-0-12-192350-1}}
* {{cite book | author=George Isac | title=Complementarity Problems | publisher=Springer | year=1992 | isbn=978-3-540-56251-1}}
* {{cite book | author=George Isac | title=Topological Methods in Complementarity Theory | publisher=Springer | year=2000 | isbn=978-0-7923-6274-6}}
* {{cite book | author=Francisco Facchinei, Jong-Shi Pang | title=Finite-Dimensional Variational Inequalities and Complementarity Problems: v.1 and v.2 | publisher=Springer | year=2003 | isbn=978-0-387-95580-3}}
* {{cite book |last=Murty |first=K. G. |title=Linear complementarity, linear and nonlinear programming |series=Sigma Series in Applied Mathematics |volume=3 |publisher=Heldermann Verlag |location=Berlin |year=1988 |pages=xlviii+629 pp. |isbn=3-88538-403-5 |url=http://ioe.engin.umich.edu/people/fac/books/murty/linear_complementarity_webbook/ |mr=949214}}
 
===Collections===
 
* {{cite book | editor=Richard Cottle, F. Giannessi, Jacques Louis Lions | title=Variational Inequalities and Complementarity Problems: Theory and Applications | publisher=John Wiley & Sons | year=1980 | isbn=978-0-471-27610-4}}
* {{cite book | editor=Michael C. Ferris, Jong-Shi Pang | title=Complementarity and Variational Problems: State of the Art | publisher=SIAM | year=1997 | isbn=978-0-89871-391-6}}
 
==External links==
*[http://www.cs.wisc.edu/cpnet/ CPNET:Complementarity Problem Net]{{Dead link|date=May 2013}}
 
[[Category:Mathematical optimization]]
[[Category:Functional analysis]]
[[Category:Topology]]
[[Category:Numerical analysis]]
 
 
{{mathanalysis-stub}}

Latest revision as of 14:08, 11 January 2015

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This can permit the medical professional know that you may be wanting him to papers your traumas, remedies and progress, or deficiency there of, to use in the court. Explain to every single medical doctor that you just observe that you will court for a accidental injuries claim. In the event you fall short to get this done, you possibly will not earn your situation.

When you use a private injury lawyer to create a profitable situation, you need to be well prepared and happy to send to almost any physical examinations deemed essential to give support for the assert of damage. In this way, you can aid your attorney accumulate voluminous proof of the damage carried out by the defendant with your case, greatly improving the chances of you economic recovery.

Consistently talk with your injury legal professional to get involved in your case. This is extremely essential as tells you what is going on and enables you to sense an element of the staff. This may minimize the stress and panic you have if you ever can be found in courtroom.

This will demonstrate the degree of your accidents, that will significantly assist your situation. Your payment could be substantially affected by how serious your injuries are. In the event you sustain actual physical personal injuries, you should picture them prior to trying to find treatment.

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Be sure you educate your manager immediately if you are hurt at work. If there's a physical injury because of vehicle accident, get in touch with an ambulance as well as the law enforcement officials the instant you can. Whenever you become wounded, by no means ensure that is stays a magic formula from your law enforcement officials or some other appropriate regulators.

In addition, in case the legal professional will never indication a retainer agreement, you must proceed to other people. Do not work together with someone who you don't truly feel is playing you. Assistance and connection are essential when employing any type of legal representative. This arrangement is the thing that outlines exactly how much their solutions can cost you, and it is essential to have that depth settled at the earliest opportunity.

If you do not completely record your injuries as soon as you is it possible to place yourself in danger of not receiving the cash that you might are entitled to in the arrangement. So, if you want to safeguard on your own, make sure you visit a doctor straight away. Constantly visit the medical facility if your damage could be the mistake of someone different.

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