Quadratic programming

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.

The quadratic programming problem with $n$ variables and $m$ constraints can be formulated as follows. Given:

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

where $x T$ denotes the vector transpose of $\mathbf {x}$ . The notation $A x \leq b$ means that every entry of the vector $A x$ is less than or equal to the corresponding entry of the vector $b$ .

A related programming problem, quadratically constrained quadratic programming, can be posed by adding quadratic constraints on the variables.

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

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

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