Convex constraint for the minimum of a vector

Question

I am solving a second-order cone programming (SOCP) problem. I had to add a constraint that checks if at least one element of the decision variable vector is lower or equal to 0, i.e., I have to add the constraint $$\min(y)\le0$$ Of course, this kind of constraint is concave and it is not compatible with SOCP.
Is there any way to write this constraint in a SOCP-like form?

Answer 1

One approach is to solve $n$ SOCP problems. First solve one copy with the constraint $y_1\le 0$, then solve another copy with the constraint $y_2 \le 0$, and so on, then taking the best solution found amongst all of these.

However, this blows up if you have many of these constraints.