Quadratic programming: Difference between revisions

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{{Short description|Solving an optimization problem with a quadratic objective function}}

'''Quadratic programming''' is a special type of mathematical [[optimization (mathematics) | optimization]] problem.

{{Expert needed|1=Mathematics|reason=Some items in this page need clarification and/or expert verification|talk=quadratic programming|date=February 2017}}

'''Quadratic programming''' ('''QP''') is the process of solving certain [[mathematical optimization]] [[optimization problem|problems]] involving [[quadratic function]]s. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear [[constrained optimization|constraints]] on the variables. Quadratic programming is a type of [[nonlinear programming]].

Quadratic programming problem can be formulated like this:

"Programming" in this context refers to a formal procedure for solving mathematical problems. This usage dates to the 1940s and is not specifically tied to the more recent notion of "computer programming." To avoid confusion, some practitioners prefer the term "optimization" — e.g., "quadratic optimization."<ref name="PrincetonCompanion">{{Citation|last=Wright|first=Stephen J.|year=2015|title=Continuous Optimization (Nonlinear and Linear Programming)|editor=Nicholas J. Higham|display-editors=etal|encyclopedia=The Princeton Companion to Applied Mathematics|pages=281–293|publisher=Princeton University Press}}</ref>

Assume x belongs to '''R'''ⁿ space. The (n x n) [[matrix]] E is [[positive semidefinite]] and h is any (n x 1) vector.

==Problem formulation==

Minimize (with respect to x)

The quadratic programming problem with {{mvar|n}} variables and {{mvar|m}} constraints can be formulated as follows.<ref>{{Cite book | last1=Nocedal | first1=Jorge | last2=Wright | first2=Stephen J. | title=Numerical Optimization | url=https://archive.org/details/numericaloptimiz00noce_639 | url-access=limited | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | edition=2nd | isbn=978-0-387-30303-1 | year=2006 | page=[https://archive.org/details/numericaloptimiz00noce_639/page/n469 449] }}.</ref>

Given:

* a [[real number|real]]-valued, {{mvar|n}}-dimensional vector {{math|'''c'''}},

* an {{math|''n''×''n''}}-dimensional real [[symmetric matrix]] {{mvar|Q}},

* an {{math|''m''×''n''}}-dimensional real [[matrix (mathematics)|matrix]] {{mvar|A}}, and

* an {{mvar|m}}-dimensional real vector {{math|'''b'''}},

the objective of quadratic programming is to find an {{mvar|n}}-dimensional vector {{math|'''x'''}}, that will

:{| cellspacing="10"

f(x) = 0.5 x' E x + h' x

|-

| minimize

| <math>\tfrac{1}{2} \mathbf{x}^\mathrm{T} Q\mathbf{x} + \mathbf{c}^\mathrm{T} \mathbf{x}</math>

|-

| subject to

| <math>A \mathbf{x} \preceq \mathbf{b},</math>

|}

where {{math|'''x'''^T}} 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).

===Constrained least squares===

with the following constraints (if there exists an answer then it satisfies these):

As a special case when {{math|''Q''}} is [[positive definite matrix|symmetric positive-definite]], the cost function reduces to least squares:

(1) A*x <= b (inequality constraint)

:{| cellspacing="10"

(2) C*x = d (equality contraint)

|-

| minimize

| <math>\tfrac{1}{2} \| R \mathbf{x} - \mathbf{d}\|^2 </math>

|-

| subject to

| <math>A \mathbf{x} \preceq \mathbf{b},</math>

|}

where {{math|1=''Q'' = ''R''^T''R''}} follows from the [[Cholesky decomposition]] of {{math|''Q''}} and {{math|1='''c''' = −''R''^T '''d'''}}. Conversely, any such [[constrained least squares]] program can be equivalently framed as a quadratic programming problem, even for a generic non-square {{math|''R''}} matrix.

===Generalizations===

If E is positive definite then f(x) is a '''convex function''' , and constraints are '''linear functions''', we have from optimization theory that for point x to be an optimum point it is necessary and sufficient that x is a Kuhn-Tucker point.

When minimizing a function {{mvar|f}} in the neighborhood of some reference point {{math|''x''₀}}, {{mvar|Q}} is set to its [[Hessian matrix]] {{math|'''H'''(''f''('''x'''₀))}} and {{math|'''c'''}} is set to its [[gradient]] {{math|∇''f''('''x'''₀)}}. A related programming problem, [[quadratically constrained quadratic program]]ming, can be posed by adding quadratic constraints on the variables.

(this article needs a lot more work..)

==Solution methods==

For general problems a variety of methods are commonly used, including

:*[[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 |url-status=live}}</ref>

:*[[Conjugate gradient method|conjugate gradient]],

:*[[Gradient projection method|gradient projection]],

:*extensions of the [[simplex algorithm]].<ref name="ioe.engin.umich" />

In the case in which {{mvar|Q}} is [[positive definite matrix|positive definite]], the problem is a special case of the more general field of [[convex optimization]].

===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 constrained problem

:<math>\text{Minimize} \quad \tfrac{1}{2} \mathbf{x}^\mathrm{T} Q\mathbf{x} + \mathbf{c}^\mathrm{T} \mathbf{x}</math>

:<math>\text{subject to} \quad E\mathbf{x} =\mathbf{d}</math>

is given by the linear system

:<math>

\begin{bmatrix}

Q & E^\top \\

E & 0

\end{bmatrix}

\begin{bmatrix} \mathbf x \\ \lambda \end{bmatrix}

=

\begin{bmatrix} -\mathbf c \\ \mathbf d \end{bmatrix}

</math>

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:

:<math>E\mathbf{x} = 0</math>

introduce a new variable {{math|'''y'''}} defined by

:<math>Z \mathbf{y} = \mathbf x</math>

where {{math|'''y'''}} has dimension of {{math|'''x'''}} minus the number of constraints. Then

:<math>E Z \mathbf{y} = \mathbf 0</math>

and if {{mvar|Z}} is chosen so that {{math|1=''EZ'' = 0}} the constraint equation will be always satisfied. Finding such {{mvar|Z}} entails finding the [[null space]] of {{mvar|E}}, which is more or less simple depending on the structure of {{mvar|E}}. Substituting into the quadratic form gives an unconstrained minimization problem:

:<math>\tfrac{1}{2} \mathbf{x}^\top Q\mathbf{x} + \mathbf{c}^\top \mathbf{x} \quad \implies \quad

\tfrac{1}{2} \mathbf{y}^\top Z^\top Q Z \mathbf{y} + \left(Z^\top \mathbf{c}\right)^\top \mathbf{y}</math>

the solution of which is given by:

:<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==

{{See also|Dual problem}}

The Lagrangian [[Dual problem|dual]] of a quadratic programming problem is also a quadratic programming problem. To see this let us focus on the case where {{math|1=''c'' = 0}} and {{mvar|Q}} is positive definite. We write the [[Lagrange multipliers|Lagrangian]] function as

:<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}}:

:<math>x^* = -Q^{-1} A^\top \lambda.</math>

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 quadratic programming problem is

:<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 ===

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 |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 n³) arithmetic operations, for a total runtime complexity of O(''L''² ''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 ===

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_i''² = ''x_i''. This constraint is equivalent to requiring that ''x_i'' is in {0,1}, that is, ''x_i'' 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>

==Solvers and scripting (programming) languages==

{| class="wikitable"

|-

!Name

!Brief info

|-

|[[AIMMS]]|| A software system for modeling and solving optimization and scheduling-type problems

|-

|[[ALGLIB]]|| Dual licensed (GPL/proprietary) numerical library (C++, .NET).

|-

|[[AMPL]]|| A popular modeling language for large-scale mathematical optimization.

|-

|[[APMonitor]]|| Modeling and optimization suite for [[Linear programming|LP]], QP, [[Nonlinear programming|NLP]], [[Integer programming|MILP]], [[Mixed Integer Nonlinear Programming|MINLP]], and [[Differential algebraic equation|DAE]] systems in MATLAB and Python.

|-

|[[Artelys Knitro]] || An Integrated Package for Nonlinear Optimization

|-

|[[CGAL]]|| An open source computational geometry package which includes a quadratic programming solver.

|-

|[[CPLEX]]|| Popular solver with an API (C, C++, Java, .Net, Python, Matlab and R). Free for academics.

|-

|[[Microsoft Excel|Excel]] Solver Function|| A nonlinear solver adjusted to spreadsheets in which function evaluations are based on the recalculating cells. Basic version available as a standard add-on for Excel.

|-

|[[General Algebraic Modeling System|GAMS]] || A high-level modeling system for mathematical optimization

|-

|[[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

|-

|[[HiGHS optimization solver|HiGHS]]|| Open-source software for solving linear programming (LP), mixed-integer programming (MIP), and convex quadratic programming (QP) models

|-

|[[IMSL Numerical Libraries|IMSL]]|| A set of mathematical and statistical functions that programmers can embed into their software applications.

|-

|[[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.

|-

|[[MATLAB]]|| A general-purpose and matrix-oriented programming-language for numerical computing. Quadratic programming in MATLAB requires the Optimization Toolbox in addition to the base MATLAB product

|-

|[[Mathematica]]|| A general-purpose programming-language for mathematics, including symbolic and numerical capabilities.

|-

|[[MOSEK]]|| A solver for large scale optimization with API for several languages (C++, Java, .Net, Matlab and Python).

|-

|[[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.

|-

|[[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.

|-

|[[R (programming language)|R]] (Fortran)||[[GNU General Public License|GPL]] licensed universal cross-platform statistical computation framework.

|-

|[[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]].

|-

|[[SuanShu numerical library|SuanShu]]|| an open-source suite of optimization algorithms to solve [[Linear programming|LP]], QP, [[SOCP]], [[Semidefinite programming|SDP]], [[Sequential quadratic programming|SQP]] in Java

|-

|[[TK Solver]]|| Mathematical modeling and problem solving software system based on a declarative, rule-based language, commercialized by Universal Technical Systems, Inc..

|-

|[[TOMLAB]]||Supports global optimization, integer programming, all types of least squares, linear, quadratic and unconstrained programming for [[MATLAB]]. TOMLAB supports solvers like [[CPLEX]], [[SNOPT]] and [[KNITRO]].

|-

|[[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==

*[[Sequential quadratic programming]]

*[[Linear programming]]

*[[Critical line method]]

==References==

{{Reflist}}

==Further reading==

* {{cite book|last1=Cottle|first1=Richard W.|last2=Pang|first2=Jong-Shi|last3=Stone|first3=Richard E.|title=The linear complementarity problem | series=Computer Science and Scientific Computing|publisher=Academic Press, Inc.|___location=Boston, MA|year=1992|pages=xxiv+762 pp|isbn=978-0-12-192350-1 | mr = 1150683 }}

* {{cite book|author-link1=Michael R. Garey|first1=Michael R.|last1=Garey|author-link2=David S. Johnson|last2=Johnson|first2=David S.| year = 1979 | title = Computers and Intractability: A Guide to the Theory of NP-Completeness |url=https://archive.org/details/computersintract0000gare|url-access=registration| publisher = W.H. Freeman | isbn = 978-0-7167-1045-5}} A6: MP2, pg.245.

* {{cite web

|first1=Nicholas I. M.|last1=Gould

|first2=Philippe L.|last2=Toint|title=A Quadratic Programming Bibliography

|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 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

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