I am trying to deal with an optimization problem with the following formulation
$$\mbox{minimize}_{x} \quad \max\limits_f \{\sum_{i=1}^N C_{i,f} \prod_{j=1}^i (1-x_{j,f}) \}, 0\leq x \leq 1$$
I don't want to solve this problem using geometric programming, since $e^x + e^{1-x}=1$ is nonconvex. Thus, my idea is to approximate the max function and the product $\prod_{j=1}^i (1-x_{j,f})$. Is there any way I can do that to approximate the original objective function to a convex function with a bounded optimality guarantee?