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#REDIRECT [[Mathematical optimization]]
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In [[mathematical optimization]], '''ordinal optimization''' is the maximization of functions taking values in a [[partially ordered set]] ("poset").{{sfn|Dietrich|Hoffman|2003}}{{sfn|Topkis|1998}}{{sfn|Singer|1997}}{{sfn|Bj&ouml;rner|Ziegler|1992}}
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Problems of ordinal optimization arise in many disciplines. Ordinal optimization has applications in the theory of [[queuing theory|queuing]] [[flow network|networks]]. In particular, [[antimatroid]]s and the "[[max-plus algebra]]" have found application in [[flow network|network analysis]] and [[queuing theory]], particularly in queuing networks and [[discrete event simulation|discrete-event systems]].{{sfn|Glasserman|Yao|1994}}{{sfn|Baccelli|Cohen|Olsder|Quadrat|1992}}{{sfn|Heidergott|Oldser|van&nbsp;der&nbsp;Woude|2006}} [[Computer science|Computer scientists]] study [[selection algorithm]]s, which are simpler than [[sorting algorithm]]s.<ref>[[Donald Knuth]]. ''[[The Art of Computer Programming]]'', Volume 3: ''Sorting and Searching'', Third Edition. Addison-Wesley, 1997. {{isbn|0-201-89685-0}}. Section 5.3.3: Minimum-Comparison Selection, pp.207&ndash;219.</ref><ref>[[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]]. ''[[Introduction to Algorithms]]'', Second Edition. MIT Press and McGraw-Hill, 2001. {{isbn|0-262-03293-7}}. Chapter 9: Medians and Order Statistics, pp.183&ndash;196. Section 14.1: Dynamic order statistics, pp.302&ndash;308.</ref> [[Statistical decision theory]] studies "selection problems" that require the identification of a "best" subpopulation or of identifying a "near best" subpopulation.<ref>[[Jean D. Gibbons|Gibbons, Jean Dickinson]]; [[Ingram Olkin|Olkin, Ingram]], and Sobel, Milton, ''Selecting and Ordering of Populations'', Wiley, (1977). (Republished as a Classic in Applied Mathematics by SIAM.)</ref><ref>{{cite book|last1=Gupta|first1=Shanti&nbsp;S.|last2=Panchapakesan|first2=S.|title=Multiple decision procedures: Theory and methodology of selecting and ranking populations|series=Wiley Series in Probability and Mathematical Statistics|publisher=John Wiley &&nbsp;Sons|location=New York|year=1979|pages=xxv+573|isbn=0-471-05177-2|mr=555416}} (Republished as a Classic in Applied Mathematics by SIAM.)</ref><ref>Santner, Thomas J., and Tamhane, A. C., ''Design of Experiments: Ranking and Selection'', M. Dekker, (1984).</ref> [[Ordered vector space|Partially ordered vector space]]s and [[Riesz space|vector lattice]]s are important in [[multiobjective optimization|optimization with multiple objectives]].<ref>{{cite book|last=Zălinescu|first=C.|title=Convex analysis in general vector spaces|publisher=World Scientific Publishing&nbsp; Co.,&nbsp;Inc|location = River Edge,&nbsp;NJ|year= 2002|pages=xx+367|isbn=981-238-067-1|mr=1921556}}</ref>

== See also ==
* [[Level of measurement]] ("Ordinal data")

== References ==
{{reflist|colwidth=30em}}

== Further reading ==
* {{wikicite|Fujishige, Satoru ''Submodular functions and optimization''. Second edition. Annals of Discrete Mathematics, 58. Elsevier B. V., Amsterdam, 2005. xiv+395 pp. {{isbn|0-444-52086-4}} <!-- MR2171629 (2006d:90098) -->|ref={{harvid|Fujishige|2005}}}}
* {{wikicite|Gondran, Michel; Minoux, Michel ''Graphs, dioids and semirings. New models and algorithms''. Operations Research/Computer Science Interfaces Series, 41. Springer, New York, 2008. xx+383 pp. {{isbn|978-0-387-75449-9}} <!-- MR2389137 (2009g:16066) -->|ref={{harvid|Gondran|Minoux|2008}}}}
* {{wikicite|Dietrich, B. L.; Hoffman, A. J. On greedy algorithms, partially ordered sets, and submodular functions. ''IBM J. Res. Dev.'' 47 (2003), no. 1, 25–30. <!-- MR1957350 (2003k:90102) -->|ref={{harvid|Dietrich|Hoffman|2003}}}}
* {{wikicite|Murota, Kazuo ''Discrete convex analysis''. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. xxii+389 pp. {{isbn|0-89871-540-7}} <!-- MR1997998 (2004f:90007) -->|ref={{harvid|Murota|2003}}}}
* {{wikicite|Topkis, Donald M. ''Supermodularity and complementarity''. Frontiers of Economic Research. Princeton University Press, Princeton, NJ, 1998. xii+272 pp. {{isbn|0-691-03244-0}} <!-- MR1614637 (99i:90024) -->|ref={{harvid|Topkis|1998}}}}
* {{wikicite|Singer, Ivan ''Abstract convex analysis''. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1997. xxii+491 pp. {{isbn|0-471-16015-6}} <!-- MR1461544 -->|ref={{harvid|Singer|1997}}}}
* {{wikicite|Björner, Anders; Ziegler, Günter M. Introduction to greedoids. ''Matroid applications'', 284–357, Encyclopedia Math. Appl., 40, Cambridge Univ. Press, Cambridge, 1992,<!-- MR1165545 (94a:05038) -->|ref={{harvid|Bj&ouml;rner|Ziegler|1992}}}}
* {{wikicite|Zimmermann, U. ''Linear and combinatorial optimization in ordered algebraic structures''. Ann. Discrete Math. 10 (1981), viii+380 pp. <!-- MR0609751 -->|ref={{harvid|Zimmermann|1981}}}}
* {{wikicite|Cuninghame-Green, Raymond ''Minimax algebra''. Lecture Notes in Economics and Mathematical Systems, 166. Springer-Verlag, Berlin-New York, 1979. xi+258 pp. {{isbn|3-540-09113-0}} <!-- MR0580321 -->|ref={{harvid|Cuninghame-Green|1979}}}}
* {{cite book|author-last1=Baccelli|author-first1=François&nbsp;Louis|author-last2=Cohen|author-first2=Guy|author-last3=Olsder|author-first3=Geert&nbsp;Jan|author-last4=Quadrat|author-first4=Jean-Pierre|title=Synchronization and linearity: An algebra for discrete event systems|series=Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics|publisher=John Wiley & Sons, Ltd.|location=Chichester|year=1992|pages=xx+489|isbn=0-471-93609-X|mr=1204266}}
* {{cite book| last1=Glasserman|first1=Paul|last2=Yao|first2=David D.|title=Monotone structure in discrete-event systems|series=Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics|publisher=John Wiley & Sons, Inc.|location=New York|year=1994|pages=xiv+297|isbn=0-471-58041-4|mr=1266839}}
* {{cite book|author-last1=Heidergott|author-first1=Bernd|author-last2=Oldser|author-first2=Geert&nbsp;Jan|author-last3=van&nbsp;der&nbsp;Woude|author-first3=Jacob|title=Max plus at work: Modeling and analysis of synchronized systems, a course on max-plus algebra and its applications|series=Princeton Series in Applied Mathematics|publisher=Princeton University Press|location=Princeton, NJ|year=2006|pages=xii+213|isbn=978-0-691-11763-8|mr=2188299}}
* [[Yu-Chi Ho|Ho, Y.C.]], Sreenivas, R., Vakili, P.,"[https://link.springer.com/content/pdf/10.1007/BF01797280.pdf Ordinal Optimization of Discrete Event Dynamic Systems]", J. of DEDS 2(2), 61-88, (1992).
* Allen, Eric, and [[Marija D. Ilić]]. ''Price-Based Commitment Decisions in the Electricity Market''. Advances in industrial control. London: Springer, 1999. {{isbn|978-1-85233-069-9}}

== External links ==
* [http://people.deas.harvard.edu/~ho/DEDS/OO/Reference/OOReference.html Annotated bibliography on ordinal optimization] by [[Yu-Chi Ho]]

{{DEFAULTSORT:Ordinal Optimization}}
[[Category:Control theory]]
[[Category:Order theory]]
[[Category:Optimization of ordered sets]]

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