QP Problem Examples

Circular paraboloid with a single constraint

Let's visualize and solve a very simple QP problem of minimizing $x_1$ and $x_2$ for an objective function in the form of a circular paraboloid and a single linear inequality constraint.

Mathematical model

$$\begin{array}{rl}\underset{x_1,x_2}{\min }& x_1^2+x_2^2\\ \text{s.t.}& -2x_1+x_2\le -1\end{array}$$

Canonical form

$$\begin{array}{rl}\mathbf{Q}& =\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]\\ \mathbf{c}& =\left[\begin{array}{c}0\\ 0\end{array}\right]\\ \mathbf{A}& =\left[\begin{array}{cc}-1& 1\end{array}\right]\\ \mathbf{b}& =\left[\begin{array}{c}-1\end{array}\right]\end{array}$$

Code

• julia
  • JuMP
  • OSQP
• python
  • CVXPY
  • OSQP

Economic dispatch

We have three power plants each producing electricity to satisfy the demand of $P_{\Sigma }=3000$ MW. Each of the power plants has a minimum and maximum capacity $P_{i_{\min }}$ and $P_{i_{\max }}$ and a quadratic cost model associated with its power output $l_i=a_i+b_i{P}_i+c_i{P}_i^2$. The capacity limits and cost coefficients are provided in the table bellow.

Mathematical model

$$\begin{array}{rl}\underset{\{P_i{\}}_{i=1}^3}{\min }& \sum _{i=1}^3{a}_i+b_i{P}_i+c_i{P}_i^2\\ \text{s.t.}& \sum _{i=1}^3{P}_i=P_{\Sigma }\\ & P_{i_{\min }}\le P_i\le P_{i_{\min }},i=1,\dots ,3\end{array}$$

OSQP form

$$\begin{array}{rl}\mathbf{P}& =\left[\begin{array}{ccc}{c}_1& 0& 0\\ 0& c_2& 0\\ 0& 0& c_3\end{array}\right]\\ \mathbf{q}& =\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]\\ \mathbf{A}& =\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]\\ \mathbf{l}& =\left[\begin{array}{c}{P}_{1_{\min }}\\ P_{2_{\min }}\\ P_{3_{\min }}\\ P_{\Sigma }\end{array}\right]\\ \mathbf{u}& =\left[\begin{array}{c}{P}_{1_{\max }}\\ P_{2_{\max }}\\ P_{3_{\max }}\\ P_{\Sigma }\end{array}\right]\end{array}$$

Code

• python - OSQP

Model Predictive Control (MPC)

Optimal control of linear systems on a finite horizon can be more often than not stated as a QP. Formulation of these problems is a topic of future chapters.