I have an optimisation problem of the form \begin{array}{cccll} \min &f(x) \\ \text{s.t.}& g_{1}(x) \leq s \\ & g_{2}(x) \leq t \\ & st \leq C \\ \end{array}
where my variables are $x \in \mathbb{R}^{n}$ and $s, t \in \mathbb{R}$, and the functions $f, g_{1}, g_{2}$ are convex in $x$, and $C$ is a constant such that the program is feasible. Is there a general approach for solving such problems? I would like to find a feasible point which is a good approximation in the sense that the value of the objective at the feasible point is close to the optimal value. I am aware of McCormick relaxation as a general technique for dealing with bilinear forms but this approach doesn't seem to return a feasible point. If it helps, it may be assumed that both $s, t$ are positive and may be bounded above by some constants $C_{s}$ and $C_{t}$.