Usage of `SecondOrderCone`: Supply value of t

[DanielDoehring]

I have a constraint of the form
$$ \Vert A \boldsymbol x + \boldsymbol b \Vert_2 \leq 1 $$ which seems to be a perfect fit for the COSMO.Constraint together with SecondOrderCone with $t=1$. However, there seems to be no way to supply the value of $t$, but only the dimensionality dimof the argument of the norm (in my case $A \boldsymbol x + \boldsymbol b$).

Unfortunately, there are no examples how to use theSecondOrderCone cone without using JuMP. I would like to avoid JuMP since it seems unnecessary overhead for my problem and also complicates (if not completely prevents) the use of higher precision types then Float64.

[migarstka]

Hi Daniel,

COSMO's native constraints are of the form $A x + b \in \mathcal{K}$
where $\mathcal{K}$ is one of the supported proper convex cones.

The second-order cone is defined as $(t,z) \in \mathcal{K}_{socp}$ such that
$| z|_2 \leq t $.

Therefore, to get your norm constraint into the right form, we define the vector $s = (t,z) = (1,z)$ and model the constraint
$| \hat{A} x + \hat{b} |_2 = z \leq 1$ as

(0, hat_A ) x + (1, hat_b) = $s \in \mathcal{K}_{socp}$.

Example

To solve the problem

$$ \begin{array}{ll} \text{maximize} & x_1 + x_2\\ \text{s.t.} & | \hat{A} x + \hat{b} |_2 \leq 1 \end{array} $$

where A = [sqrt(2)/4, 0; 0, \sqrt(2)/4] and b = [sqrt(2)/4; sqrt(2)/4], I would write the constraints natively like this:

using COSMO, SparseArrays, LinearAlgebra, Test

q = [-1; -1.];
P = spzeros(2, 2);
A = [sqrt(2)/4 0; 0 sqrt(2)/4];
b = [√2/4; √2/4]
# We model the norm constraint using `COSMO.SecondOrderCone` as the convex set:
Aa = [0. 0; A]
ba = [1; b]
constraint1 = COSMO.Constraint(Aa, ba, COSMO.SecondOrderCone(3));

# Next, we define the settings object, the model and then assemble everything:
settings = COSMO.Settings(verbose=true);
model = COSMO.Model();
assemble!(model, P, q, constraint1, settings = settings);
res = COSMO.optimize!(model);

The solution should be $x_1=x_2 = 1$ if I am not mistaken.