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where {{math|'''x'''<sup>T</sup>}} denotes the vector [[transpose]] of {{math|'''x'''}}, and the notation {{math|''A'''''x''' ⪯ '''b'''}} means that every entry of the vector {{math|''A'''''x'''}} is less than or equal to the corresponding entry of the vector {{math|'''b'''}} (component-wise inequality).
===
As a special case when {{math|''Q''}} is [[positive definite matrix|symmetric positive-definite]], the cost function reduces to least squares:
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| <math>A \mathbf{x} \preceq \mathbf{b},</math>
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where {{math|1=''Q'' = ''R''<sup>T</sup>''R''}} follows from the [[Cholesky decomposition]] of {{math|''Q''}} and {{math|1='''c''' = −''R''<sup>T</sup> '''d'''}}. Conversely, any such [[constrained least squares]] program can be equivalently framed as a
===Generalizations===
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:*[[interior point method|interior point]],
:*[[active set]],<ref name="ioe.engin.umich">{{cite book|last=Murty|first=Katta G.|title=Linear complementarity, linear and nonlinear programming|series=Sigma Series in Applied Mathematics|volume=3|publisher=Heldermann Verlag|___location=Berlin|year=1988|pages=xlviii+629 pp|isbn=978-3-88538-403-8|url=http://ioe.engin.umich.edu/people/fac/books/murty/linear_complementarity_webbook/|mr=949214|url-status=dead|archive-url=https://web.archive.org/web/20100401043940/http://ioe.engin.umich.edu/people/fac/books/murty/linear_complementarity_webbook/|archive-date=2010-04-01}}</ref>
:*[[Augmented Lagrangian method|augmented Lagrangian]],<ref>{{cite journal | first1 = F. | last1 = Delbos | first2 = J.Ch. | last2 = Gilbert | year = 2005 | title = Global linear convergence of an augmented Lagrangian algorithm for solving convex quadratic optimization problems | journal = Journal of Convex Analysis | volume = 12 | pages = 45–69 |url=http://www.heldermann-verlag.de/jca/jca12/jca1203_b.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.heldermann-verlag.de/jca/jca12/jca1203_b.pdf |archive-date=2022-10-09
:*[[Conjugate gradient method|conjugate gradient]],
:*[[Gradient projection method|gradient projection]],
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===Equality constraints===
Quadratic programming is particularly simple when {{mvar|Q}} is [[positive definite matrix|positive definite]] and there are only equality constraints; specifically, the solution process is linear. By using [[Lagrange multipliers]] and seeking the extremum of the Lagrangian, it may be readily shown that the solution to the equality
:<math>\text{Minimize} \quad \tfrac{1}{2} \mathbf{x}^\mathrm{T} Q\mathbf{x} + \mathbf{c}^\mathrm{T} \mathbf{x}</math>
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where {{math|λ}} is a set of Lagrange multipliers which come out of the solution alongside {{math|'''x'''}}.
The easiest means of approaching this system is direct solution (for example, [[LU factorization]]), which for small problems is very practical. For large problems, the system poses some unusual difficulties, most notably that the problem is never positive definite (even if {{mvar|Q}} is), making it potentially very difficult to find a good numeric approach, and there are many approaches to choose from dependent on the problem.
If the constraints don't couple the variables too tightly, a relatively simple attack is to change the variables so that constraints are unconditionally satisfied. For example, suppose {{math|1='''d''' = 0}} (generalizing to nonzero is straightforward). Looking at the constraint equations:
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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''<sup>T</sup>''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==
{{See also|Dual problem}}
The Lagrangian [[Dual problem|dual]] of a
:<math>L(x,\lambda) = \tfrac{1}{2} x^\top Qx + \lambda^\top (Ax-b). </math>
Defining the (Lagrangian) dual function {{math|''g''(λ)}} as <math>g(\lambda) = \inf_{x} L(x,\lambda) </math>, we find an [[infimum]] of {{mvar|L}}, using <math>\nabla_{x} L(x,\lambda)=0</math> and positive-definiteness of {{mvar|Q}}:
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Hence the dual function is
:<math>g(\lambda) = -\tfrac{1}{2} \lambda^\top AQ^{-1}A^\top \lambda - \lambda^\top b,</math>
and so the Lagrangian dual of the
:<math>\text{maximize}_{\lambda\geq 0} \quad -\tfrac{1}{2} \lambda^\top AQ^{-1} A^\top \lambda - \lambda^\top b.</math>
Besides the Lagrangian duality theory, there are other duality pairings (e.g. [[Wolfe duality|Wolfe]], etc.).
==Run-time complexity==
=== Convex quadratic programming ===
Ye and Tse<ref>{{Cite journal |last1=Ye |first1=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> present a polynomial-time algorithm, which extends [[Karmarkar's algorithm]] from linear programming to convex quadratic programming. On a system with ''n'' variables and ''L'' input bits, their algorithm requires O(L n) iterations, each of which can be done using O(L n<sup>3</sup>) arithmetic operations, for a total runtime complexity of O(''L''<sup>2</sup> ''n''<sup>4</sup>).
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''<sup>3.67</sup> * log ''n'') arithmetic operations.
=== Non-convex quadratic programming ===
If {{mvar|Q}} is indefinite, (so the problem is non-convex) then the problem is [[NP-hard]].<ref>{{cite journal | last = Sahni | first = S. | title = Computationally related problems | journal = SIAM Journal on Computing | volume = 3 | issue = 4 | pages = 262–279 | year = 1974 | doi=10.1137/0203021| url = http://www.cise.ufl.edu/~sahni/papers/comp.pdf | citeseerx = 10.1.1.145.8685 }}</ref> A simple way to see this is to consider the non-convex quadratic constraint ''x<sub>i</sub>''<sup>2</sup> = ''x<sub>i</sub>''. This constraint is equivalent to requiring that ''x<sub>i</sub>'' is in {0,1}, that is, ''x<sub>i</sub>'' is a binary integer variable. Therefore, such constraints can be used to model any [[Integer programming|integer program]] with binary variables, which is known to be NP-hard.
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 ==
There are some situations where one or more elements of the vector {{math|'''x'''}} will need to take on [[integer]] values. This leads to the formulation of a mixed-integer quadratic programming (MIQP) problem.<ref>{{Cite journal|last=Lazimy|first=Rafael|date=1982-12-01|title=Mixed-integer quadratic programming|journal=Mathematical Programming| language=en| volume=22| issue=1| pages=332–349| doi=10.1007/BF01581047| s2cid=8456219|issn=1436-4646}}</ref> Applications of MIQP include [[water resources]]<ref>{{Cite journal|last1=Propato Marco|last2=Uber James G.|date=2004-07-01|title=Booster System Design Using Mixed-Integer Quadratic Programming|journal=Journal of Water Resources Planning and Management|volume=130|issue=4|pages=348–352|doi=10.1061/(ASCE)0733-9496(2004)130:4(348)}}</ref> and the [[Tracking error#Index fund creation|construction of index funds]].<ref>{{Cite book|last1=Cornuéjols|first1=Gérard|url=https://www.cambridge.org/core/books/optimization-methods-in-finance/8A4996C5DB2006224E4D983B5BC95E3B|title=Optimization Methods in Finance|last2=Peña|first2=Javier|last3=Tütüncü|first3=Reha|publisher=Cambridge University Press|year=2018|isbn=9781107297340|edition=2nd|___location=Cambridge, UK|pages=167–168}}</ref>
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|[[GNU Octave]]|| A free (its licence is [[GPL]]v3) general-purpose and matrix-oriented programming-language for numerical computing, similar to MATLAB. Quadratic programming in GNU Octave is available via its [https://www.gnu.org/software/octave/doc/interpreter/Quadratic-Programming.html qp] command
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|[[HiGHS optimization solver|HiGHS]]|| Open-source software for solving linear programming (LP), mixed-integer programming (MIP), and convex quadratic programming (QP) models
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|[[IMSL Numerical Libraries|IMSL]]|| A set of mathematical and statistical functions that programmers can embed into their software applications.
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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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|[[NAG Numerical Library]]|| A collection of mathematical and statistical routines developed by the [[Numerical Algorithms Group]] for multiple programming languages (C, C++, Fortran, Visual Basic, Java and C#) and packages (MATLAB, Excel, R, LabVIEW). The Optimization chapter of the NAG Library includes routines for quadratic programming problems with both sparse and non-sparse linear constraint matrices, together with routines for the optimization of linear, nonlinear, sums of squares of linear or nonlinear functions with nonlinear, bounded or no constraints. The NAG Library has routines for both local and global optimization, and for continuous or integer problems.
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|[[Python (programming language)|Python]]||High-level programming language with bindings for most available solvers. Quadratic programming is available via the [https://pypi.org/project/qpsolvers/ solve_qp] function or by calling a specific solver directly.
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|[[SAS System|SAS]]/OR|| A suite of solvers for Linear, Integer, Nonlinear, Derivative-Free, Network, Combinatorial and Constraint Optimization; the [[Algebraic modeling language]] OPTMODEL; and a variety of vertical solutions aimed at specific problems/markets, all of which are fully integrated with the [[SAS System]].
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|[[
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|[[TK Solver]]|| Mathematical modeling and problem solving software system based on a declarative, rule-based language, commercialized by Universal Technical Systems, Inc..
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|[[FICO Xpress|XPRESS]]||Solver for large-scale linear programs, quadratic programs, general nonlinear and mixed-integer programs. Has API for several programming languages, also has a modelling language Mosel and works with AMPL, [[General Algebraic Modeling System|GAMS]]. Free for academic use.
|}
== Extensions ==
'''Polynomial optimization'''<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]]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
}}
==External links==
*[http://www.numerical.rl.ac.uk/qp/qp.html A page about
*[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
{{Mathematical programming}}
{{Optimization algorithms}}
{{Authority control}}
[[Category:Optimization algorithms and methods]]
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