# Quadratic programming

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**Quadratic programming** (QP) is the process of solving a special type of mathematical optimization problem—specifically, a (linearly constrained) quadratic optimization problem, that is, the problem of optimizing (minimizing or maximizing) a quadratic function of several variables subject to linear constraints on these variables. Quadratic programming is a particular type of nonlinear programming.

## Problem formulation[edit]

The quadratic programming problem with n variables and m constraints can be formulated as follows.^{[1]}
Given:

- a real-valued, n-dimensional vector
**c**, - an
*n*×*n*-dimensional real symmetric matrix Q, - an
*m*×*n*-dimensional real matrix A, and - an m-dimensional real vector
**b**,

the objective of quadratic programming is to find an n-dimensional vector **x**, that will

where **x**^{T} denotes the vector transpose of . The notation *A***x** ⪯ **b** means that every entry of the vector *A***x** is less than or equal to the corresponding entry of the vector **b** (componentwise inequality).

### Least squares[edit]

As a special case when Q is symmetric positive definite, the cost function reduces to least squares:

where **Q** = **R**^{T}**R** follows from the Cholesky decomposition of **Q** and **c** = -**R**^{T} **d**. Conversely, any such constrained least squares program can be equivalently framed as a QP, even for generic non-square **R** matrix.

### Generalizations[edit]

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

## Solution methods[edit]

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

- interior point,
- active set,
^{[2]} - augmented Lagrangian,
^{[3]} - conjugate gradient,
- gradient projection,
- extensions of the simplex algorithm.
^{[2]}

In the case in which *Q* is positive definite, the problem is a special case of the more general field of convex optimization.

### Equality constraints[edit]

Quadratic programming is particularly simple when *Q* is 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

is given by the linear system

where is a set of Lagrange multipliers which come out of the solution alongside .

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 is), making it potentially very difficult to find a good numeric approach, and there are many approaches to choose from dependent on the problem.^{[4]}

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 (generalizing to nonzero is straightforward). Looking at the constraint equations:

introduce a new variable defined by

where has dimension of minus the number of constraints. Then

and if is chosen so that the constraint equation will be always satisfied. Finding such entails finding the null space of , which is more or less simple depending on the structure of . Substituting into the quadratic form gives an unconstrained minimization problem:

the solution of which is given by:

Under certain conditions on , the reduced matrix will be positive definite. It is possible to write a variation on the conjugate gradient method which avoids the explicit calculation of .^{[5]}

## Lagrangian duality[edit]

The Lagrangian dual of a QP is also a QP. To see that let us focus on the case where and Q is positive definite. We write the Lagrangian function as

Defining the (Lagrangian) dual function as , we find an infimum of , using and positive-definiteness of Q:

Hence the dual function is

and so the Lagrangian dual of the QP is

Besides the Lagrangian duality theory, there are other duality pairings (e.g. Wolfe, etc.).

## Complexity[edit]

For positive definite *Q*, the ellipsoid method solves the problem in (weakly) polynomial time.^{[6]} If, on the other hand, *Q* is indefinite, then the problem is NP-hard.^{[7]} In fact, even if *Q* has only one negative eigenvalue, the problem is (strongly) NP-hard.^{[8]}

## Integer constraints[edit]

There are some situations where one or more elements of the vector will need to take on integer values. This leads to the formulation of a mixed-integer quadratic programming (MIQP) problem.^{[9]} Applications of MIQP include water resources^{[10]} and the construction of index funds.^{[11]}

## Solvers and scripting (programming) languages[edit]

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 LP, QP, NLP, MILP, MINLP, and DAE systems in MATLAB and Python. |

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. |

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. |

GAMS | A high-level modeling system for mathematical optimization |

Gurobi | Solver with parallel algorithms for large-scale linear programs, quadratic programs and mixed-integer programs. Free for academic use. |

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 |

Artelys Knitro | An Integrated Package for Nonlinear Optimization |

Maple | General-purpose programming language for mathematics. Solving a quadratic problem in Maple is accomplished via its 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. |

GNU Octave | A free (its licence is GPLv3) general-purpose and matrix-oriented programming-language for numerical computing, similar to MATLAB. Quadratic programming in GNU Octave is available via its qp command |

R (Fortran) | GPL licensed universal cross-platform statistical computation framework. |

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. |

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 Gurobi, CPLEX, SNOPT and KNITRO. |

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, GAMS. Free for academic use. |

## See also[edit]

## References[edit]

**^**Nocedal, Jorge; Wright, Stephen J. (2006).*Numerical Optimization*(2nd ed.). Berlin, New York: Springer-Verlag. p. 449. ISBN 978-0-387-30303-1..- ^
^{a}^{b}Murty, Katta G. (1988).*Linear complementarity, linear and nonlinear programming*. Sigma Series in Applied Mathematics.**3**. Berlin: Heldermann Verlag. pp. xlviii+629 pp. ISBN 978-3-88538-403-8. MR 0949214. Archived from the original on 2010-04-01. **^**Delbos, F.; Gilbert, J.Ch. (2005). "Global linear convergence of an augmented Lagrangian algorithm for solving convex quadratic optimization problems" (PDF).*Journal of Convex Analysis*.**12**: 45–69.**^**Google search.**^**Gould, Nicholas I. M.; Hribar, Mary E.; Nocedal, Jorge (April 2001). "On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization".*SIAM J. Sci. Comput*.**23**(4): 1376–1395. CiteSeerX 10.1.1.129.7555. doi:10.1137/S1064827598345667.**^**Kozlov, M. K.; S. P. Tarasov; Leonid G. Khachiyan (1979). "[Polynomial solvability of convex quadratic programming]".*Doklady Akademii Nauk SSSR*.**248**: 1049–1051. Translated in:*Soviet Mathematics - Doklady*.**20**: 1108–1111. Missing or empty`|title=`

(help)**^**Sahni, S. (1974). "Computationally related problems" (PDF).*SIAM Journal on Computing*.**3**(4): 262–279. CiteSeerX 10.1.1.145.8685. doi:10.1137/0203021.**^**Pardalos, Panos M.; Vavasis, Stephen A. (1991). "Quadratic programming with one negative eigenvalue is (strongly) NP-hard".*Journal of Global Optimization*.**1**(1): 15–22. doi:10.1007/bf00120662. S2CID 12602885.**^**Lazimy, Rafael (1982-12-01). "Mixed-integer quadratic programming".*Mathematical Programming*.**22**(1): 332–349. doi:10.1007/BF01581047. ISSN 1436-4646. S2CID 8456219.**^**Propato Marco; Uber James G. (2004-07-01). "Booster System Design Using Mixed-Integer Quadratic Programming".*Journal of Water Resources Planning and Management*.**130**(4): 348–352. doi:10.1061/(ASCE)0733-9496(2004)130:4(348).**^**Cornuéjols, Gérard; Peña, Javier; Tütüncü, Reha (2018).*Optimization Methods in Finance*(2nd ed.). Cambridge, UK: Cambridge University Press. pp. 167–168. ISBN 9781107297340.

## Further reading[edit]

- Cottle, Richard W.; Pang, Jong-Shi; Stone, Richard E. (1992).
*The linear complementarity problem*. Computer Science and Scientific Computing. Boston, MA: Academic Press, Inc. pp. xxiv+762 pp. ISBN 978-0-12-192350-1. MR 1150683. - Garey, Michael R.; Johnson, David S. (1979).
*Computers and Intractability: A Guide to the Theory of NP-Completeness*. W.H. Freeman. ISBN 978-0-7167-1045-5. A6: MP2, pg.245. - Gould, Nicholas I. M.; Toint, Philippe L. (2000). "A Quadratic Programming Bibliography" (PDF). RAL Numerical Analysis Group Internal Report 2000-1.