Solving an optimization problem with a quadratic objective function
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"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."[1]
Problem formulation
The quadratic programming problem with n variables and m constraints can be formulated as follows.[2]
Given:
the objective of quadratic programming is to find an n-dimensional vector x, that will
minimize
subject to
where xT denotes the vector transpose of x, and the notation Ax ⪯ b means that every entry of the vector Ax is less than or equal to the corresponding entry of the vector b (component-wise inequality).
where Q = RTR follows from the Cholesky decomposition of Q and c = −RTd. Conversely, any such constrained least squares program can be equivalently framed as a quadratic programming problem, even for a generic non-square R matrix.
Generalizations
When minimizing a function f in the neighborhood of some reference point x0, Q is set to its Hessian matrixH(f(x0)) and c is set to its gradient∇f(x0). A related programming problem, quadratically constrained quadratic programming, can be posed by adding quadratic constraints on the variables.
Solution methods
For general problems a variety of methods are commonly used, including
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 λ is a set of Lagrange multipliers which come out of the solution alongside 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 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 d = 0 (generalizing to nonzero is straightforward). Looking at the constraint equations:
introduce a new variable y defined by
where y has dimension of x minus the number of constraints. Then
and if Z is chosen so that EZ = 0 the constraint equation will be always satisfied. Finding such Z entails finding the null space of E, which is more or less simple depending on the structure of E. Substituting into the quadratic form gives an unconstrained minimization problem:
the solution of which is given by:
Under certain conditions on Q, the reduced matrix ZTQZ will be positive definite. It is possible to write a variation on the conjugate gradient method which avoids the explicit calculation of Z.[5]
The Lagrangian dual of a quadratic programming problem is also a quadratic programming problem. To see this let us focus on the case where c = 0 and Q is positive definite. We write the Lagrangian function as
Defining the (Lagrangian) dual function g(λ) as , we find an infimum of L, using and positive-definiteness of Q:
Hence the dual function is
and so the Lagrangian dual of the quadratic programming problem is
Besides the Lagrangian duality theory, there are other duality pairings (e.g. Wolfe, etc.).
Ye and Tse[7] 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(L2n4).
Kapoor and Vaidya[8] present another algorithm, which requires O(L * log L* n3.67 * log n) arithmetic operations.
Non-convex quadratic programming
If Q is indefinite, (so the problem is non-convex) then the problem is NP-hard.[9] A simple way to see this is to consider the non-convex quadratic constraint xi2 = xi. This constraint is equivalent to requiring that xi is in {0,1}, that is, xi is a binary integer variable. Therefore, such constraints can be used to model any 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 Q has only one negative eigenvalue, the problem is (strongly) NP-hard.[10]
Moreover, finding a KKT point of a non-convex quadratic program is CLS-hard.[11]
Mixed-integer quadratic programming
There are some situations where one or more elements of the vector x will need to take on integer values. This leads to the formulation of a mixed-integer quadratic programming (MIQP) problem.[12] Applications of MIQP include water resources[13] and the construction of index funds.[14]
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.
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
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
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.
High-level programming language with bindings for most available solvers. Quadratic programming is available via the solve_qp function or by calling a specific solver directly.
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.
Mathematical modeling and problem solving software system based on a declarative, rule-based language, commercialized by Universal Technical Systems, Inc..
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.
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.
Extensions
Polynomial optimization[15] is a more general framework, in which the constraints can be polynomial functions of any degree, not only 2.
^Wright, Stephen J. (2015), "Continuous Optimization (Nonlinear and Linear Programming)", in Nicholas J. Higham; et al. (eds.), The Princeton Companion to Applied Mathematics, Princeton University Press, pp. 281–293
^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. CiteSeerX10.1.1.129.7555. 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. {{cite journal}}: Missing or empty |title= (help)
^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. S2CID12602885.
^Fearnley, John; Goldberg, Paul W.; Hollender, Alexandros; Savani, Rahul (2023). "The Complexity of Computing KKT Solutions of Quadratic Programs". arXiv:2311.13738 [cs.CC].
^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).
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. ISBN978-0-12-192350-1. MR1150683.