General question
An SOCP constraint is given by:
$$ \| A_i \mathbf{x} + b_i\| \leq \mathbf{c}_i^T \mathbf{x} + d_i.$$
I have the following constraint:
$$ \| A_i \mathbf{x} \| \geq d_i.$$
Is it possible to apply some kind of trick to convexify this constraint?
The specific case for which convexification might be possible?
In my problem, the constraint aims at avoiding subsurface flight of a vehicle in orbit by dictating that the norm of the vehicle's radius vector must always be larger than the radius of the planet, e.g.:
$$\|\mathbf{r} \| \geq R,$$
where $\mathbf{r}$ is the radius vector and $R$ is the planet's radius. Is it possible to apply some trick and make this constraint convex in this case?
PS:
I am aware of question 619727, but that is a slightly different case
There is no mathematical trick to convexify a constraint like that. It's the complement of a convex set. That's about as "far" from convex as you can get with a connected set.
That said, it might be possible for you to find a way to modify your model so that it is more readily handled using convex optimization. You might look at the work of Dr. Behcet Acikmese at the University of Texas. He's written papers on the "convexification" of certain convex models involved in guidance, navigation, and control problems.
Here is a PDF of one of his conference papers, which he co-wrote with Lars Blackmore. It might be a good place to start. I make no guarantees that it will apply to your application. Here is a longer list of papers by Blackmore et al on the use of convex optimization for guidance, navigation, and control.