This is a Linear Program but quadratic now.

\begin{align} \min_{x}\ &\qty(\frac{1}{2}) 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}\qty(c^{T}x) \\ &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 \(\text{var}\qty(c^{T}x)\). So, writing in terms of a QP:

minimize ¯cT x + yxT Σx subject to Gx  h, Ax = b