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| In mathematical [[optimization (mathematics)|optimization theory]], the '''linear complementarity problem (LCP)''' arises frequently in [[computational mechanics]] and encompasses the well-known [[quadratic programming]] as a special case. It was proposed by Cottle and [[George Dantzig|Dantzig]] {{nowrap|in 1968.<ref name="Murty88"/><ref name="CPS92"/><ref>R. W. Cottle and [[G. B. Dantzig]]. Complementary pivot theory of mathematical programming. ''Linear Algebra and its Applications'', 1:103-125, 1968.</ref>}}
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| == Formulation ==
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| Given a real matrix '''M''' and vector '''q''', the linear complementarity problem seeks vectors '''z''' and '''w''' which satisfy the following constraints:
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| * <math>{w} = {Mz} + {q} \,</math>
| | Anyone invest loads of funds on things like controls and for memory cards, appear on-line for a secondhand release. Occasionally a store will probably are out of used-game hardware, which could be inexpensive. Make sure you look at the web-based seller's feedback before making the purchase so this whether you are having what you covered.<br><br>Beginning nearly enough rocks to get another developer. Don''t waste some of the gems from any way on rush-building anything, as if they can save you the group you are going so that it will eventually obtain enough freely available extra gems to grab that extra builder without even cost. Particularly, you may can get free gallstones for clearing obstructions favor rocks and trees, subsequent you clear them out and about they come back and also you may re-clear these to get more rocks.<br><br>To take pleasure from unlimited points, resources, gold coins or gems, you needs to download the clash of clans compromise tool by clicking on his or her button. If you beloved this short article and you would like to receive much more data regarding [http://circuspartypanama.com clash of clans trainer] kindly pay a visit to our own web-page. [http://www.google.co.uk/search?hl=en&gl=us&tbm=nws&q=Depending&gs_l=news Depending] regarding the operating system that the using, you will should run the downloaded file as administrator. Deliver the log in ID and choose the device. After this, you are ought to enter the number pointing to gems or coins that you prefer to get.<br><br>Principally clash of clans get into tool no survey builds believe in among people. Society is just definitely powered by look pressure, one of the most powerful forces during the planet. Like long as peer demands utilizes its power to gain good, clash of clans hack tool no stare at will have its space in community.<br><br>Hold out for game of this particular season editions of worthwhile titles. These regularly come out per the four seasons or higher after your current initial headline, but include things like a lot of all down-loadable and extra happy which was released appearing in steps once the first headline. These video titles supply a good more bang for the buck.<br><br>A person are are the proud possessor of an ANY easily transportable device that runs located on iOS or android basically a touchscreen tablet computer or a smart phone, then you definitely would need to have already been sensitive of the revolution taking place right now a world of mobile high-def game "The Clash In Clans", and you will probably be in demand of a conflict of families totally free of charge jewels compromise because good deal more gems, elixir and sterling silver are needed seriously to acquire every battle.<br><br>Certainly individuals who produced this unique Crack Clash of Families are true fans related with the sport themselves, and this is exactly the things ensures the potency at our alternative, because some of us needed to do that it ourselves. |
| * <math>{w} \ge 0, {z} \ge 0\,</math> (that is, each component of these two vectors is non-negative)
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| * <math>{w}_i {z}_i = 0\,</math> for all i. (The [[Complementarity theory|complementarity]] condition)
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| A sufficient condition for existence and uniqueness of a solution to this problem is that '''M''' be [[Symmetric matrix|symmetric]] [[Positive-definite matrix|positive-definite]].
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| The vector <math>{w}\,</math> is a [[slack variable]], and so is generally discarded after <math>{z}\,</math> is found.{{Citation needed|date=March 2012}} As such, the problem can also be formulated as:
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| * <math>{Mz}+{q} \ge {0}\,</math>
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| * <math>{z} \ge {0}\,</math>
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| * <math>{z}^{\mathrm{T}}({Mz}+{q}) = 0\,</math> (the complementarity condition)
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| ==Convex quadratic-minimization: Minimum conditions==
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| Finding a solution to the linear complementarity problem is associated with minimizing the quadratic function
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| : <math>f({z}) = {z}^{\mathrm{T}}({Mz}+{q})\,</math>
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| subject to the constraints
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| : <math>{Mz}+{q} \ge {0}\,</math>
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| : <math>{z} \ge {0}\,</math>
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| These constraints ensure that ''f'' is always non-negative. The minimum of ''f'' is 0 at '''z''' if and only if '''z''' solves the linear complementarity problem.
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| If '''M''' is [[Positive-definite matrix|positive definite]], any algorithm for solving (strictly) convex [[Quadratic programming|QPs]] can solve the LCP. Specially designed basis-exchange pivoting algorithms, such as [[Lemke's algorithm]] and a variant of the [[simplex algorithm|simplex algorithm of Dantzig]] have been used for decades. Besides having polynomial time complexity, interior-point methods are also effective in practice.
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| Also, a quadratic-programming problem stated as minimize <math>f({x})={c}^T{x}+\frac{1}{2}{x}^T{Qx}\,</math> subject to <math>{Ax} \geq {b} \,</math> as well as <math>{x} \ge {0}\,</math> with ''Q'' symmetric
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| is the same as solving the LCP with
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| * <math>{q} = \left[\begin{array}{c}{c}\\-{b}\end{array}\right]\,</math>
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| * <math>{M} = \left[\begin{array}{cc} {Q} & -{A}^{T}\\ {A} & 0\end{array}\right]\,</math>
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| This is because the [[Karush–Kuhn–Tucker]] conditions of the QP problem can be written as:
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| * <math>{v} = {Q} {x} - {A}^{T} {\lambda} + {c}\,</math>
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| * <math>{s} = {A} {x} - {b}\,</math>
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| * <math>{x}, {\lambda}, {v}, {s} \ge {0}\,</math>
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| * <math>{x}^{T}{v} + {\lambda}^{T}{s} = {0}\,</math>
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| ...being <math> {v} \,</math> the Lagrange multipliers on the non-negativity constraints,<math> {\lambda} \,</math> the <!-- Lagrange -->multipliers on the inequality constraints, and <math> {s} \,</math> the slack variables for the inequality constraints. The fourth condition derives from the complementarity of each group of variables (<math>{x}, {s}\,</math>) with its set of KKT vectors (optimal Lagrange multipliers) being (<math>{v}, {\lambda}\,</math>).
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| In that case,
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| : <math>{z} = \left[\begin{array}{c}{x}\\ {\lambda}\end{array}\right]\,</math>
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| : <math>{w} = \left[\begin{array}{c}{v}\\ {s}\end{array}\right]\,</math>
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| If the non-negativity constraint on the <math>{x}\,</math> is relaxed, the dimensionality of the LCP problem can be reduced to the number of the inequalities, as long as <math>{Q}\,</math> is non-singular (which is guaranteed if it is [[Positive-definite matrix|positive definite]]). The multipliers <math>{v}\,</math> are no longer present, and the first KKT conditions can be rewritten as:
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| * <math>{Q} {x} = {A}^{T} {\lambda} - {c}\,</math>
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| or:
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| : <math> {x} = {Q}^{-1}({A}^{T} {\lambda} - {c})\,</math>
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| pre-multiplying the two sides by <math>{A}\,</math> and subtracting <math>{b}\,</math> we obtain:
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| : <math> {A} {x} - {b} = {A} {Q}^{-1}({A}^{T} {\lambda} - {c}) -{b} \,</math>
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| The left side, due to the second KKT condition, is <math>{s}\,</math>. Substituting and reordering:
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| : <math> {s} = ({A} {Q}^{-1} {A}^{T}) {\lambda} + (- {A} {Q}^{-1} {c} - {b} )\,</math>
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| Calling now <math> {M} \,\overset{\underset{\mathrm{def}}{}}{=}\, ({A} {Q}^{-1} {A}^{T})\,</math> and <math> {q} \,\overset{\underset{\mathrm{def}}{}}{=}\, (- {A} {Q}^{-1} {c} - {b})\,</math> we have an LCP, due to the relation of complementarity between the slack variables <math>{s}\,</math> and their Lagrange multipliers <math>{\lambda}\,</math>. Once we solve it, we may obtain the value of <math>{x}\,</math> from <math>{\lambda}\,</math> through the first KKT condition.
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| Finally, it is also possible to handle additional equality constraints:
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| : <math>{A}_{eq}{x} = {b}_{eq} \,</math>
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| This introduces a vector of Lagrange multipliers <math>{\mu}\,</math>, with the same dimension as <math>{b}_{eq}\,</math>.
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| It is easy to verify that the <math>{M}\,</math> and <math>{q}\,</math> for the LCP system <math> {s} = {M} {\lambda} + {q} \,</math> are now expressed as:
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| : <math> {M} ~\overset{\underset{\mathrm{def}}{}}{=}~ \left(\left[\begin{array}{cc}{A} & {0}\end{array}\right] \left[\begin{array}{cc} {Q} & {A}_{eq}^{T}\\ -{A}_{eq} & {0}\end{array}\right]^{-1} \left[\begin{array}{cc}{A}^{T} \\ {0}\end{array}\right]\right)\,</math>
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| : <math> {q} ~\overset{\underset{\mathrm{def}}{}}{=}~ \left(- \left[\begin{array}{cc}{A} & {0}\end{array}\right] \left[\begin{array}{cc} {Q} & {A}_{eq}^{T}\\ -{A}_{eq} & {0}\end{array}\right]^{-1} \left[\begin{array}{c}{c}\\ {b}_{eq}\end{array}\right] - {b}\right)\,</math>
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| From <math>{\lambda}\,</math> we can now recover the values of both <math>{x}\,</math> and the Lagrange multiplier of equalities <math>{\mu}\,</math>:
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| <math>\left[\begin{array}{c}{x}\\ {\mu}\end{array}\right] = \left[\begin{array}{cc} {Q} & {A}_{eq}^{T}\\ -{A}_{eq} & {0}\end{array}\right]^{-1} \left[\begin{array}{c} {A}^{T}{\lambda}-{c}\\-{b}_{eq}\end{array}\right] \,</math>
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| In fact, most QP solvers work on the LCP formulation, including the [[interior point method]], principal / complementarity pivoting, and [[active set]] methods.<ref name="Murty88">{{harvtxt|Murty|1988}}</ref><ref name="CPS92">{{harvtxt|Cottle|Pang|Stone|1992}}</ref> LCP problems can be solved also by the [[criss-cross algorithm]],<ref>{{harvtxt|Fukuda|Namiki|1994}}</ref><ref>{{harvtxt|Fukuda|Terlaky|1997}}</ref><ref name="HRT">{{cite journal|first1=D. |last1=den Hertog|first2=C.|last2=Roos|first3=T.|last3=Terlaky|title=The linear complementarity problem, sufficient matrices, and the criss-cross method|journal=Linear Algebra and its Applications|volume=187|date=1 July 1993|pages=1–14|doi=10.1016/0024-3795(93)90124-7|url=http://www.sciencedirect.com/science/article/pii/0024379593901247|<!-- ref=harv -->|url=http://arno.uvt.nl/show.cgi?fid=72943|format=pdf}}</ref><ref name="CIsufficient">{{cite journal|first1=Zsolt|last1=Csizmadia|first2=Tibor|last2=Illés|title=New criss-cross type algorithms for linear complementarity problems with sufficient matrices|journal=Optimization Methods and Software|volume=21|year=2006|number=2|pages=247–266|doi=10.1080/10556780500095009|
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| url=http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf|format=pdf|url2=http://www.tandfonline.com/doi/abs/10.1080/10556780500095009|eprint=http://www.tandfonline.com/doi/pdf/10.1080/10556780500095009|mr=2195759|<!-- ref=harv -->}}</ref> conversely, for linear complementarity problems, the criss-cross algorithm terminates finitely only if the matrix is a sufficient matrix.<ref name="HRT"/><ref name="CIsufficient"/> A [[sufficient matrix]] is a generalization both of a [[positive-definite matrix]] and of a [[P-matrix]], whose [[principal minor]]s are each positive.<ref name="HRT"/><ref name="CIsufficient"/><ref>{{cite journal|last1=Cottle|first1=R. W.|authorlink1=Richard W. Cottle|last2=Pang|first2=J.-S.|last3=Venkateswaran|first3=V.|title=Sufficient matrices and the linear complementarity problem|journal=Linear Algebra and its Applications|volume=114–115|year=1989|pages=231–249|doi=10.1016/0024-3795(89)90463-1|url=http://www.sciencedirect.com/science/article/pii/0024379589904631|month=March–April|mr=986877|ref=harv}}</ref>
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| Such LCPs can be solved when they are formulated abstractly using [[oriented matroid|oriented-matroid]] theory.<ref name="Todd" >{{harvtxt|Todd|1985|}}</ref><ref>{{harvtxt|Terlaky|Zhang|1993}}: {{cite journal|last1=Terlaky|first1=Tamás|<!-- authorlink1=Tamás Terlaky -->|last2=Zhang|first2=Shu Zhong|title=Pivot rules for linear programming: A Survey on recent theoretical developments|issue=Degeneracy in optimization problems|journal=Annals of Operations Research|volume=46–47|year=1993|issue=1|pages=203–233|doi=10.1007/BF02096264|mr=1260019|id= {{citeseerx|10.1.1.36.7658}} |publisher=Springer Netherlands|issn=0254-5330|ref=harv}}</ref><ref>{{cite book|last=Björner|first=Anders|last2=Las Vergnas|author2-link=Michel Las Vergnas|first2=Michel|last3=Sturmfels|first3=Bernd|authorlink3=Bernd Sturmfels|last4=White|first4=Neil|last5=Ziegler|first5=Günter|authorlink5=Günter M. Ziegler|title=Oriented Matroids|chapter=10 Linear programming|publisher=Cambridge University Press|year=1999|isbn=978-0-521-77750-6|url=http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511586507|pages=417–479|doi=10.1017/CBO9780511586507|MR=1744046}}</ref>
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| == See also ==
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| *[[Complementarity theory]]
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| *[[Physics engine]] Impulse/constraint type physics engines for games use this approach.
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| *[[Contact dynamics]] Contact dynamics with the nonsmooth approach
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| ==Notes==
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| {{Reflist}}
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| == References ==
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| * {{cite book|last1=Cottle|first1=Richard W.|last2=Pang|first2=Jong-Shi|last3=Stone|first3=Richard E.|title=The linear complementarity problem | series=Computer Science and Scientific Computing|publisher=Academic Press, Inc.|location=Boston, MA|year=1992|pages=xxiv+762 pp.|isbn=0-12-192350-9|mr=1150683|ref=harv}}
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| *{{cite journal|last1=Cottle|first1=R. W.|authorlink1=Richard W. Cottle|last2=Pang|first2=J.-S.|last3=Venkateswaran|first3=V.|title=Sufficient matrices and the linear complementarity problem|journal=Linear Algebra and its Applications|volume=114–115|year=1989|pages=231–249|doi=10.1016/0024-3795(89)90463-1|url=http://www.sciencedirect.com/science/article/pii/0024379589904631|month=March–April|mr=986877|ref=harv}}
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| * {{cite journal|first1=Zsolt|last1=Csizmadia|first2=Tibor|last2=Illés|title=New criss-cross type algorithms for linear complementarity problems with sufficient matrices|journal=Optimization Methods and Software|volume=21|year=2006|number=2|pages=247–266|doi=10.1080/10556780500095009|
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| url=http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf|format=pdf|url2=http://www.tandfonline.com/doi/abs/10.1080/10556780500095009|eprint=http://www.tandfonline.com/doi/pdf/10.1080/10556780500095009|mr=2195759|<!-- ref=harv -->}}
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| * {{cite journal|last1=Fukuda|first1=Komei|authorlink1=Komei Fukuda|last2=Namiki|first2=Makoto|title=On extremal behaviors of Murty's least index method|journal=Mathematical Programming|date=March 1994|pages=365–370|volume=64|issue=1|doi=10.1007/BF01582581|ref=harv|mr=1286455}}
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| * {{cite journal|first1=D. |last1=den Hertog|first2=C.|last2=Roos|first3=T.|last3=Terlaky|title=The linear complementarity problem, sufficient matrices, and the criss-cross method|journal=Linear Algebra and its Applications|volume=187|date=1 July 1993|pages=1–14|doi=10.1016/0024-3795(93)90124-7|url=http://www.sciencedirect.com/science/article/pii/0024379593901247|<!-- ref=harv -->|url=http://arno.uvt.nl/show.cgi?fid=72943|format=pdf}}
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| * {{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|id=[http://www-personal.umich.edu/~murty/ Updated and free PDF version at Katta G. Murty's website]|ref=harv}}
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| * {{cite journal|first1=Komei|last1=Fukuda|<!-- authorlink1=Komei Fukuda -->|first2=Tamás|last2=Terlaky|<!-- authorlink2=Tamás Terlaky -->|title=Criss-cross methods: A fresh view on pivot algorithms |doi=10.1007/BF02614325|journal=Mathematical Programming: Series B|volume=79
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| |issue=1—3|pages=369–395|issue=Papers from the 16th International Symposium on Mathematical Programming held in Lausanne, 1997|editors=Thomas M. Liebling and Dominique de Werra|publisher=North-Holland Publishing Co. |location=Amsterdam|year=1997|doi=10.1016/S0025-5610(97)00062-2|mr=1464775|ref=harv|id=[http://www.cas.mcmaster.ca/~terlaky/files/crisscross.ps Postscript preprint]}}
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| *{{cite journal|last=Todd|first=Michael J.|authorlink=Michael J. Todd (mathematician)|title=Linear and quadratic programming in oriented matroids|journal=Journal of Combinatorial Theory|series=Series B|volume=39|year=1985|issue=2|pages=105–133|mr=811116|doi=10.1016/0095-8956(85)90042-5|ref=harv}}
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| ==Further reading==
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| * R. W. Cottle and [[G. B. Dantzig]]. Complementary pivot theory of mathematical programming. ''Linear Algebra and its Applications'', 1:103-125, 1968.
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| * {{cite journal|last1=Terlaky|first1=Tamás|<!-- authorlink1=Tamás Terlaky -->|last2=Zhang|first2=Shu Zhong|title=Pivot rules for linear programming: A Survey on recent theoretical developments|issue=Degeneracy in optimization problems|journal=Annals of Operations Research|volume=46–47|year=1993|issue=1|pages=203–233|doi=10.1007/BF02096264|mr=1260019|id = {{citeseerx|10.1.1.36.7658}} |publisher=Springer Netherlands|issn=0254-5330|ref=harv}}
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| == External links ==
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| * [http://www1.american.edu/econ/gaussres/optimize/quadprog.src LCPSolve] — A simple procedure in GAUSS to solve a linear complementarity problem
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| * [http://www.openopt.org/LCP LCPSolve.py] — A Python/NumPy implementation of LCPSolve is part of [[OpenOpt]] since its release 0.32
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| * [[Siconos]]/Numerics open-source GPL implementation in C of Lemke's algorithm and other methods to solve LCPs and MLCPs
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| {{Mathematical programming}}
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| [[Category:Linear algebra]]
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| [[Category:Mathematical optimization]]
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