Criss-cross algorithm: Difference between revisions

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The criss-cross algorithm is simpler than the simplex algorithm, because the criss-cross algorithm only has one phase. Its pivoting rules are similar to the [[Bland's rule|least-index pivoting rule of Bland]].<ref name="Bland">

{{cite journal|title=New finite pivoting rules for the simplex method|first=Robert G.|last=Bland|journal=Mathematics of Operations Research|volume=2|number=2|date=May 1977|pages=103–107|doi=10.1287/moor.2.2.103|jstor=3689647|mr=459599}}</ref> Bland's rule uses only [[sign function|sign]]s of coefficients rather than their [[real number#Axiomatic approach|(real-number) order]] when deciding eligible pivots. Bland's rule selects an entering variables by comparing values of reduced costs, using the real-number ordering of the eligible pivots.<ref name="Bland"/><ref>Bland's rule is also related to an earlier least-index rule, which was proposed by Katta&nbsp;G. Murty for the [[linear complementarity problem]], according to {{harvtxt|Fukuda|Namiki|1994}}.</ref> Unlike Bland's rule, the criss-cross algorithm is "purely combinatorial", selecting an entering variable and a leaving variable by considering only the signs of coefficients rather than their real-number ordering.<ref name="FukudaTerlaky"/><ref name="TerlakyZhang"/> The criss-cross algorithm has been applied to furnish constructive proofs of basic results in [[linear algebra]], such as <!-- [[Steinitz's theorem|Steinitz's lemma]], --> the [[Farkas lemma|lemma of Farkas]]<!-- , [[Weyl's theorem]] on the finite generation of [[convex polytope]]s by linear inequalities ([[halfspace]]s), and the [[Krein–Milman theorem|Minkowski's theorem]] on [[extreme point]]s -->.<ref name="KT91" >{{harvtxt|Klafszky|Terlaky|1991}}</ref>

While most simplex variants are monotonic in the objective (strictly in the non-degenerate case), most variants of the criss-cross algorithm lack a monotone merit function which can be a disadvantage in practice.

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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|last1=Björner|first1=Anders|last2=Las Vergnas|first2=Michel|author2-link=Michel Las Vergnas|last3=Sturmfels|first3=Bernd|author-link3=Bernd Sturmfels|last4=White|first4=Neil|last5=Ziegler|first5=Günter|author-link5=Günter M. Ziegler|title=Oriented Matroids|chapter=10 Linear programming|publisher=Cambridge University Press|year=1999|isbn=978-0-521-77750-6|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|date=1 July 1993|pages=1–14|url=httphttps://core.ac.uk/download/pdf/6714737.pdf|doi=10.1016/0024-3795(93)90124-7|doi-access=free}}</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<!--|eprint=http://www.tandfonline.com/doi/pdf/10.1080/10556780500095009-->|mr=2195759|s2cid=24418835|access-date=30 August 2011|archive-date=23 September 2015|archive-url=https://web.archive.org/web/20150923211403/http://www.cs.elte.hu/opres/orr/download/ORR03_1.pdf|url-status=dead}}</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&nbsp;matrix]] is a generalization both of a [[positive-definite matrix]] and of a [[P-matrix]], whose [[principal&nbsp;minor]]s are each positive.<ref name="HRT"/><ref name="CIsufficient"/><ref>{{cite journal|last1=Cottle|first1=R. W.|author-link1=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|date=March–April 1989|pages=231–249|doi=10.1016/0024-3795(89)90463-1|mr=986877|doi-access=free}}</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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* {{cite journal|last1=Fukuda|first1=Komei|author-link1=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|number=1|doi=10.1007/BF01582581|mr=1286455|s2cid=21476636}}

* {{cite journal|first1=Komei|last1=Fukuda| author-link1=Komei Fukuda |first2=Tamás|last2=Terlaky| author-link2=Tamás Terlaky |title=Criss-cross methods: A fresh view on pivot algorithms |journal=Mathematical Programming, Series B|volume=79|pages=369–395|issue=Papers from the 16th International Symposium on Mathematical Programming held in Lausanne, 1997, number 1–3 |editor1-first=Thomas M.|editor1-last=Liebling|editor2-first=Dominique|editor2-last=de Werra|year=1997|doi=10.1007/BF02614325|mr=1464775|id=[http://www.cas.mcmaster.ca/~terlaky/files/crisscross.ps Postscript preprint]|citeseerx=10.1.1.36.9373|s2cid=2794181}}

* {{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|url=httphttps://core.ac.uk/download/pdf/6714737.pdf|doi=10.1016/0024-3795(93)90124-7|mr=1221693|doi-access=free}}

* {{<!-- citation -->cite journal|title=The finite criss-cross method for hyperbolic programming|journal=European Journal of Operational Research|volume=114|number=1|

pages=198–214|year=1999<!-- |issn=0377-2217 -->|doi=10.1016/S0377-2217(98)00049-6|url=http://www.sciencedirect.com/science/article/B6VCT-3W3DFHB-M/2/4b0e2fcfc2a71e8c14c61640b32e805a