Questions on quadratic programming, the optimization of a quadratic objective function subject to affine constraints.

Quadratic Programming (QP) is a special case of convex-optimization where a quadratic function (called the *objective function*) of one or several variables is optimised subject to only affine constraints. Compared to linear-programming (LP), only the objective function is different.

For $n$ variables and $m$ constraints, the objective is find an $n\times1$ vector $\bf x$ under the following conditions. \begin{align}\min\quad&\frac12{\bf x}^\top Q{\bf x}+{\bf c}^\top{\bf x}\\\text{s.t.}\quad&A{\bf x}\preceq{\bf b}\end{align} where, in the real numbers,

$\bf c$ is an $n\times1$ vector

$\bf b$ is an $m\times1$ vector

$Q$ is an $n\times n$ symmetric matrix

$A$ is an $m\times n$ matrix.

Common methods to solve them include the augmented Lagrangian and conjugate gradients.

Reference: Wikipedia - Quadratic Programming