This is a Linear Program but quadratic now.
\begin{align} \min_{x}\ &\left(\frac{1}{2}\right) x^{T} P x + q^{T} x + r \\ s.t.\ &Gx \preceq h \\ & Ax = b \end{align}
We want P \in S_{+}^{n}, so PSD. So its convex quadratic. Examples Least Squares Obviously least-squares is a basic Quadratic Program
\begin{equation} \norm{A x - b}^{2}_{2} \end{equation}
Linear Program with Random Cost Consider a linear program with stochastic cost c with mean \bar{c} and covariance \Sigma. Hence, a Linear Program objective c^{T}x is a random variable with mean \bar{c}^{T}x and variance x^{T} \Sigma x. Thus the risk-average decision problem is:
\begin{align} &\min \mathbb{E}c^{T}x + \gamma \text{var}\left(c^{T}x\right) \\ &s.t.\ G x \preceq h, Ax = b \end{align}
where \gamma > 0 is risk-aversion: trading off expected cost \mathbb{E} c^{T}x and variance \text{var}\left(c^{T}x\right). So, writing in terms of a QP: minimize ¯cT x + yxT Σx subject to Gx h, Ax = b