Quadratic programming: Difference between revisions

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:<math>Z^\top Q Z \mathbf{y} = -Z^\top \mathbf{c}</math>

Under certain conditions on {{mvar|Q}}, the reduced matrix {{math|''Z''^T''QZ''}} will be positive definite. It is possible to write a variation on the [[conjugate gradient method]] which avoids the explicit calculation of {{mvar|Z}}.<ref>{{Cite journal | last1 = Gould| first1 = Nicholas I. M.| last2 = Hribar| first2 = Mary E.| last3 = Nocedal| first3 = Jorge|date=April 2001| title = On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization| journal = SIAM J. Sci. Comput.| pages = 1376–1395| volume = 23| issue = 4| citeseerx = 10.1.1.129.7555| doi = 10.1137/S1064827598345667| bibcode = 2001SJSC...23.1376G}}</ref>

==Lagrangian duality==

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For [[positive-definite matrix|positive definite]] {{mvar|Q}}, when the problem is convex, the [[ellipsoid method]] solves the problem in (weakly) [[polynomial time]].<ref>{{cite journal| last=Kozlov | first=M. K. |author2=S. P. Tarasov | author3-link=Leonid Khachiyan |author3=Leonid G. Khachiyan | year=1979 | title=[Polynomial solvability of convex quadratic programming] | journal=[[Doklady Akademii Nauk SSSR]] | volume=248 | pages=1049–1051}} Translated in: {{cite journal| journal=Soviet Mathematics - Doklady | volume=20 | pages=1108–1111}}</ref>

Ye and Tse<ref>{{Cite journal |lastlast1=Ye |firstfirst1=Yinyu |last2=Tse |first2=Edison |date=1989-05-01 |title=An extension of Karmarkar's projective algorithm for convex quadratic programming |url=https://doi.org/10.1007/BF01587086 |journal=Mathematical Programming |language=en |volume=44 |issue=1 |pages=157–179 |doi=10.1007/BF01587086 |s2cid=35753865 |issn=1436-4646|url-access=subscription }}</ref> andpresent Kapoora andpolynomial-time Vaidya<ref>{{Citealgorithm, journalwhich |last=Kapoorextends |first=S[[Karmarkar's |last2=Vaidyaalgorithm]] |first2=Pfrom Mlinear |date=1986-11-01programming |title=Fast algorithms forto convex quadratic programming and multicommodity flows |url=https://dl.acm.org/doi/10.1145/12130.12145 |journal=ProceedingsOn ofa thesystem eighteenthwith annual''n'' ACMvariables symposiumand on''L'' Theoryinput ofbits, computingtheir |series=STOCalgorithm '86requires |___location=NewO(L York,n) NYiterations, USAeach |publisher=Associationof forwhich Computingcan Machinerybe |pages=147–159done |doi=10.1145/12130.12145using O(L |isbn=978-0-89791-193-1}}n<sup>3</refsup>) present polynomial-timearithmetic algorithmsoperations, whichfor extend [[Karmarkar'sa algorithm]]total fromruntime linearcomplexity programmingof toO(''L''² convex quadratic programming''n''⁴).

Kapoor and Vaidya<ref>{{Cite book |last1=Kapoor |first1=S |last2=Vaidya |first2=P M |chapter=Fast algorithms for convex quadratic programming and multicommodity flows |date=1986-11-01 |title=Proceedings of the eighteenth annual ACM symposium on Theory of computing - STOC '86 |chapter-url=https://dl.acm.org/doi/10.1145/12130.12145 |___location=New York, NY, USA |publisher=Association for Computing Machinery |pages=147–159 |doi=10.1145/12130.12145 |isbn=978-0-89791-193-1|s2cid=18108815 }}</ref> present another algorithm, which requires O(''L'' * log ''L'' ''* n''^3.67 * log ''n'') arithmetic operations.

=== Non-convex quadratic programming ===

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Moreover, these non-convex problems might have several stationary points and local minima. In fact, even if {{mvar|Q}} has only one negative [[eigenvalue]], the problem is (strongly) [[NP-hard]].<ref>{{cite journal | title = Quadratic programming with one negative eigenvalue is (strongly) NP-hard | first1 = Panos M. | last1 = Pardalos | first2 = Stephen A. | last2 = Vavasis | journal = Journal of Global Optimization | volume = 1 | issue = 1 | year = 1991 | pages = 15–22 | doi=10.1007/bf00120662| s2cid = 12602885 }}</ref>

Moreover, finding a KKT point of a non-convex quadratic program is CLS-hard.<ref>{{Cite arXiv |eprint=2311.13738 |last1=Fearnley |first1=John |last2=Goldberg |first2=Paul W. |last3=Hollender |first3=Alexandros |last4=Savani |first4=Rahul |title=The Complexity of Computing KKT Solutions of Quadratic Programs |date=2023 |class=cs.CC }}</ref>

== Mixed-integer quadratic programming ==

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|-

|[[IPOPT]]|| IPOPT (Interior Point OPTimizer) is a software package for large-scale nonlinear optimization.

|-

|[[Julia (programming language)|Julia]]

|A high-level programming language with notable solving package being [https://jump.dev/JuMP.jl/stable/ JuMP]

|-

|[[Maple (software)|Maple]]|| General-purpose programming language for mathematics. Solving a quadratic problem in Maple is accomplished via its [http://www.maplesoft.com/support/help/Maple/view.aspx?path=Optimization/QPSolve QPSolve] command.

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== Extensions ==

'''Polynomial optimizaitonoptimization'''<ref>{{Citation |last=Tuy |first=Hoang |title=Polynomial Optimization |date=2016 |url=https://doi.org/10.1007/978-3-319-31484-6_12 |work=Convex Analysis and Global Optimization |pages=435–452 |editor-last=Tuy |editor-first=Hoang |access-date=2023-12-16 |series=Springer Optimization and Its Applications |volume=110 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-31484-6_12 |isbn=978-3-319-31484-6|url-access=subscription }}</ref> is a more general framework, in which the constraints can be [[polynomial function]]<nowiki/>s of any degree, not only 2.

==See also==

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|publisher=RAL Numerical Analysis Group Internal Report 2000-1

|url=ftp://ftp.numerical.rl.ac.uk/pub/qpbook/qp.pdf

|archive-url=https://web.archive.org/web/20170705054523/ftp://ftp.numerical.rl.ac.uk/pub/qpbook/qp.pdf|archive-date=2017-07-05|url-status=dead|date=2000

|date=2000

}}

==External links==

*[http://www.numerical.rl.ac.uk/qp/qp.html A page about quadratic programming] {{Webarchive|url=https://web.archive.org/web/20110607212051/http://www.numerical.rl.ac.uk/qp/qp.html |date=2011-06-07 }}

*[https://neos-guide.org/content/quadratic-programming NEOS Optimization Guide: Quadratic Programming]

*[https://www.math.uh.edu/~rohop/fall_06/Chapter3.pdf Quadratic Programming] {{Webarchive|url=https://web.archive.org/web/20230408093722/https://www.math.uh.edu/~rohop/fall_06/Chapter3.pdf |date=2023-04-08 }}

*[https://or.stackexchange.com/q/989/2576 Cubic programming and beyond], in Operations Research stack exchange