Editing Quadratic programming (section)

==Problem formulation==
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"
|-
| 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'''<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).

===Constrained least squares===

As a special case when {{math|''Q''}} is [[positive definite matrix|symmetric positive-definite]], the cost function reduces to least squares:
:{| cellspacing="10"
|-
| 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''<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 quadratic programming problem, even for a generic non-square {{math|''R''}} matrix.

===Generalizations===

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