Criss-cross algorithm: Difference between revisions

Revision as of 19:42, 7 May 2021

Revision as of 02:42, 12 May 2021

Line 3:

<!-- {{Context|date=May 2012}} -->

[[File:Unitcube.svg|thumb|right|alt=A three-dimensional cube|The criss-cross algorithm visits all&nbsp;8 corners of the [[Klee–Minty cube]] in the worst case. It visits&nbsp;3 additional corners on&nbsp;average. The Klee–Minty cube is a perturbation of the cube shown here.]]

In [[optimization (mathematics)|mathematical optimization]], the '''criss-cross algorithm''' is any of a family of [[algorithm]]s for [[linear programming]]. Variants of the criss-cross algorithm also solve more general problems with [[linear programming|linear inequality constraints]] and [[nonlinear programming|nonlinear]] [[optimization (mathematics)|objective functions]]; there are criss-cross algorithms for [[linear-fractional programming]] problems,<ref name="LF99Hyperbolic">{{harvtxt|Illés|Szirmai|Terlaky|1999}}</ref><ref name="Bibl" >{{cite journal|first=I.&nbsp;M.|last=Stancu-Minasian|title=A sixth bibliography of fractional programming|journal=Optimization|volume=55|number=4|date=August 2006|pages=405–428|doi=10.1080/02331930600819613|mr=2258634|s2cid=62199788}}</ref> [[quadratic programming|quadratic-programming]] problems, and [[linear complementarity problem]]s.<ref name="FukudaTerlaky" >{{harvtxt|Fukuda|Terlaky|1997}}</ref>

Like the [[simplex algorithm]] of [[George Dantzig|George B. Dantzig]], the criss-cross algorithm is not a [[time complexity|polynomial-time algorithm]] for linear programming. Both algorithms visit all&nbsp;2^''D''&nbsp;corners of a (perturbed) [[unit cube|cube]] in dimension&nbsp;''D'', the [[Klee–Minty cube]] (after [[Victor Klee]] and [[George J. Minty]]), in the [[worst-case complexity|worst case]].<ref name="Roos" >{{harvtxt|Roos|1990}}</ref><ref name="KleeMinty"/> However, when it is started at a random corner, the criss-cross algorithm [[Average-case complexity|on&nbsp;average]] visits only&nbsp;''D'' additional corners.<ref name="FTNamiki"/><ref name="FukudaNamiki"/><ref name="Borgwardt">The simplex algorithm takes on average&nbsp;''D'' steps for a cube. {{harvtxt|Borgwardt|1987}}: {{cite book|last=Borgwardt|first=Karl-Heinz|title=The simplex method: A probabilistic analysis|series=Algorithms and Combinatorics (Study and Research Texts)|volume=1|publisher=Springer-Verlag|___location=Berlin|year=1987|pages=xii+268|isbn=978-3-540-17096-9|mr=868467}}</ref> Thus, for the three-dimensional cube, the algorithm visits all&nbsp;8 corners in the worst case and exactly&nbsp;3 additional corners on&nbsp;average.

Line 38:

===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|lastlast1=Björner|firstfirst1=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=http://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}}</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}}</ref> The criss-cross algorithm has been adapted also for [[linear-fractional programming]].<ref name="LF99Hyperbolic"/><ref name="Bibl"/>

===Vertex enumeration===

Line 78:

* {{cite journal |first1=David |last1=Avis |first2=Komei |last2=Fukuda |author-link2=Komei Fukuda |author-link1=David Avis |title=A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra|journal=[[Discrete and Computational Geometry]] |volume=8 |date=December 1992 |pages=295–313 |doi=10.1007/BF02293050 |issue=ACM Symposium on Computational Geometry (North Conway, NH, 1991) number 1 |mr=1174359|doi-access=free }}

* {{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}}

* {{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=http://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

|first1=Tibor|last1=Illés|first2=Ákos|last2=Szirmai|first3=Tamás|last3=Terlaky|zbl=0953.90055|id=[http://www.cas.mcmaster.ca/~terlaky/files/dut-twi-96-103.ps.gz Postscript preprint]}}

*{{cite journal|first1=Emil|last1=Klafszky|first2=Tamás|last2=Terlaky|title=The role of pivoting in proving some fundamental theorems of linear algebra|journal=Linear Algebra and Its Applications|volume=151|date=June 1991|pages=97–118|doi=10.1016/0024-3795(91)90356-2|url=http://www.cas.mcmaster.ca/~terlaky/files/pivot-la.ps|format=postscript|mr=1102142|access-date=4 August 2011|archive-url=https://web.archive.org/web/20110927061207/http://www.cas.mcmaster.ca/~terlaky/files/pivot-la.ps|archive-date=27 September 2011|url-status=dead|doi-access=free}}

* {{cite journal|last=Roos|first=C.|title=An exponential example for Terlaky's pivoting rule for the criss-cross simplex method|journal=Mathematical Programming|volume=46|year=1990|number=1|series=Series A|pages=79–84|doi=10.1007/BF01585729|mr=1045573|s2cid=33463483}}<!-- Google scholar reported no free versions -->

* {{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|mr=798939}}<!-- Google scholar reported no free versions -->

* {{cite journal|last=Terlaky|first=Tamás|author-link=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}}<!-- Google scholar reported no free versions -->

* {{cite journal|last1=Terlaky|first1=Tamás| author-link1=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|pages=203–233 |doi=10.1007/BF02096264|mr=1260019 |citeseerx = 10.1.1.36.7658 |s2cid=6058077| orig-year = 1991 |issn=0254-5330}}

* {{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}}<!-- Google scholar reported no free versions -->