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{{about|an [[algorithm]] for [[optimization (mathematics)|mathematical optimization]]|the naming of [[analytical chemistry|chemicals]]|crisscross method}}
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===Other optimization problems with linear constraints===
There are variants of the criss-cross algorithm for linear programming, for [[quadratic programming]], and for the [[linear complementarity problem|linear-complementarity problem]] with "sufficient matrices";<ref name="FukudaTerlaky"/><ref name="FTNamiki"/><ref name="FukudaNamikiLCP" >{{harvtxt|Fukuda|Namiki|1994|}}</ref><ref name="OMBook" >{{cite book|last=Björner|first=Anders|last2=Las Vergnas|first2=Michel|author2-link=Michel Las Vergnas|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><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|month=1 July|year=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|
url=http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf|format=pdf|url=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> The criss-cross algorithm has been adapted also for [[linear-fractional programming]].<ref name="LF99Hyperbolic"/><ref name="Bibl"/>
===Vertex enumeration===
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==References==
* {{cite journal |url=http://www.springerlink.com/content/m7440v7p3440757u/ |first1=David |last1=Avis |first2=Komei |last2=Fukuda |authorlink2=Komei Fukuda |authorlink1=David Avis |title=A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra|journal=[[Discrete and Computational Geometry]] |volume=8
* {{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|
url=http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf|format=pdf|url=http://www.tandfonline.com/doi/abs/10.1080/10556780500095009<!--|eprint=http://www.tandfonline.com/doi/pdf/10.1080/10556780500095009--> |mr=2195759|ref=harv}}
* {{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|year=1994|month=March|pages=365–370|volume=64|number=1|doi=10.1007/BF01582581|ref=harv|mr=1286455}}
* {{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
* {{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|month=1 July|year=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|mr=1221693}}
* {{<!-- citation -->cite journal|title=The finite criss-cross method for hyperbolic programming|journal=European Journal of Operational Research|volume=114|number=1|
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* {{cite journal|last=Terlaky|first=T.|title=A convergent criss-cross method|journal=Optimization: A Journal of Mathematical Programming and Operations Research|volume=16|year=1985|number=5|pages=683–690|issn=0233-1934|doi=10.1080/02331938508843067|ref=harv|mr=798939|<!-- Google scholar reported no free versions -->}}
* {{cite journal|last=Terlaky|first=Tamás|authorlink=Tamás Terlaky|title=A finite crisscross method for oriented matroids|volume=42|year=1987|number=3|pages=319–327|journal=Journal of Combinatorial Theory|series=Series B|issn=0095-8956|doi=10.1016/0095-8956(87)90049-9|mr=888684|ref=harv|<!-- Google scholar reported no free versions -->}}
* {{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, number 1 |journal=Annals of Operations Research|volume=46–47|year=1993
* {{cite journal|last=Wang|first=Zhe Min|title=A finite conformal-elimination free algorithm over oriented matroid programming|journal=Chinese Annals of Mathematics (Shuxue Niankan B Ji)|series=Series B|volume=8|year=1987|number=1|pages=120–125|issn=0252-9599|mr=886756|ref=harv|<!-- Google scholar reported no free versions -->}}
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