I have an optimization problem of the form
$$\min_{\boldsymbol{x\in \mathbb{R}^{n+1}_+}} f(x_0,x_1,\cdots,x_n)\\ \text{s.t. } \sum_{i=0}^{n} x_i \le C_1\\ g(x_0) \le C_2 $$
the objective $f(x_0,\cdots,x_n)$ is a convex function of the argumetns, however the inequality constraint $g(x_0)$ is not convex, unless I further assume $x_0\le C_3$. All the variables and parameters (given constants $C_1,C_2,C_3$) are positive real numbers. The function $g(x_0)$ is a convex function if $x_0\le C_3$ and is a concave function if $x_0\ge C_3$ and is always increasing. Is there anyway that I can turn this problem into a convex optimization problem?
Thank you!