Approximating $ {L}_{0} $ Norm Minimization with Non Linear Convex Inequality Constraints using Reweighted $ {L}_{1} $ Minimization
I think testing it practically rather that its theoretical inspection might help more. Therefore, I tried to solve the above mentioned problem using CVX toolbox in Matlab.

cvx_begin;
   variable x(1000);
   minimise(norm(x,1));
   subject to
      x.^2&lt;=150; % f(x) here, is x^2
      0&lt;=x&lt;10;
cvx_end


I think the answer to your question depends on your function f(x), and sometimes even if there is a solution based on convexification of the stated problem, I do not see any guarantees that the answer is must be right. I mean using l1 norm as proxy for l0 norm (as what CS suggests) in this case, might be wrong and bring no fruit. Be careful with blind application of l1 norm minimization method. This great lecture from Prof. Baranuik called "Compressive nonSensing" might be help you more on this regard:
You might find it here:
http://www.norbertwiener.umd.edu/FFT/2015/