Power Cone Example

In this example we show how to model optimization problems with 3-dimensional power cone constraints. The power cone is defined as

\[`\mathcal{K}_{pow}(\alpha) = \{(x, y, z) \mid x^\alpha y^{(1-\alpha)} \geq |z|, (x,y) \geq 0 \} \]

with $\alpha \in (0,1)$.

We will solve the following optimization problem:

\[\begin{array}{ll} \text{maximize} & x_1^{0.6} y^{0.4} + x_2^{0.1}\\[2ex] \text{subject to} & \begin{array}{rl} (x_1, y, x_2) &\ge 0 \\ x_1 + 2y + 3x_2 &= 3 \end{array} \end{array}\]

which is equivalent to

\[\begin{array}{ll} \text{maximize} & z_1 + z_2\\[2ex] \text{subject to} & \begin{array}{rl} (x1, y, z1) &\in~\mathcal{K}_{pow}(0.6) \\ (x2, 1, z2) &\in~\mathcal{K}_{pow}(0.1) \\ x_1 + 2y + 3x_2 &=~3. \end{array} \end{array}\]