Basic SOCP Example

Suppose that we want to solve the following 2-dimensional optimization problem:

\[\begin{array}{ll} \text{minimize} & x_2^2\\[2ex] \text{subject to} & \left\|\begin{pmatrix} 2x_1 \\ x_2 \end{pmatrix} - \begin{pmatrix} 2 \\ 2 \end{pmatrix}\right\|_2 \le 1 \end{array}\]

We will show how to solve this problem through each of Clarabel's supported languages and interfaces.

Objective function

The Clarabel solver's default configuration expects problem data in the form $\frac{1}{2}x^\top P x + q^\top x$. We therefore define the objective function data as

\[P = 2 \cdot \begin{bmatrix} 0 & 0 \\ 0 & 1\end{bmatrix} \textrm{~~~and~~~} q = \begin{bmatrix} 0 \\ 0\end{bmatrix}.\]

Constraints

The solver's default configuration expects constraints in the form $Ax + s = b$, where $s \in \mathcal{K}$ for some composite cone $\mathcal{K}$. We have a single constraint on the 2-norm of a vector, so we rewrite

\[\left\|\begin{pmatrix} 2x_1 \\ x_2 \end{pmatrix} - \begin{pmatrix} 2 \\ 2 \end{pmatrix}\right\|_2 \le 1 \quad \Longleftrightarrow \quad \begin{pmatrix} 1 \\ 2x_1 - 2\\ x_2 - 2 \end{pmatrix} \in \mathcal{K}_{SOC}\]

which puts our constraint in the form $b - Ax \in \mathcal{K}_{SOC}$.