Quadratic programming

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

minimize	${\displaystyle {\tfrac {1}{2}}\|R\mathbf {x} -\mathbf {d} \|^{2}}$
subject to	${\displaystyle A\mathbf {x} \preceq \mathbf {b} ,}$

where Q = R^TR 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 quadratic programming problem, even for a generic non-square R matrix.

When minimizing a function f in the neighborhood of some reference point x₀, Q is set to its Hessian matrix H(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.

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

${\displaystyle {\text{Minimize}}\quad {\tfrac {1}{2}}\mathbf {x} ^{\mathrm {T} }Q\mathbf {x} +\mathbf {c} ^{\mathrm {T} }\mathbf {x} }$

${\displaystyle {\text{subject to}}\quad E\mathbf {x} =\mathbf {d} }$

is given by the linear system

${\displaystyle {\begin{bmatrix}Q&E^{\top }\\E&0\end{bmatrix}}{\begin{bmatrix}\mathbf {x} \\\lambda \end{bmatrix}}={\begin{bmatrix}-\mathbf {c} \\\mathbf {d} \end{bmatrix}}}$

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:

${\displaystyle E\mathbf {x} =0}$

introduce a new variable y defined by

${\displaystyle Z\mathbf {y} =\mathbf {x} }$

where y has dimension of x minus the number of constraints. Then

${\displaystyle EZ\mathbf {y} =\mathbf {0} }$

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:

${\displaystyle {\tfrac {1}{2}}\mathbf {x} ^{\top }Q\mathbf {x} +\mathbf {c} ^{\top }\mathbf {x} \quad \implies \quad {\tfrac {1}{2}}\mathbf {y} ^{\top }Z^{\top }QZ\mathbf {y} +\left(Z^{\top }\mathbf {c} \right)^{\top }\mathbf {y} }$

the solution of which is given by:

${\displaystyle Z^{\top }QZ\mathbf {y} =-Z^{\top }\mathbf {c} }$

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