I have a concave function minimization problem as follows: $f(x)$ = $|c_1+c_2\cdot x|\cdot x$, with constants $c_1$, $c_2 \in \mathbb{R}^{n}$ and $\vec{x}\in\mathbb{R}^{n}_{++}$, such that $c_1\prec 0$, $c_2\succ0$, and $c_1+c_2\cdot x\prec 0$. I am trying to minimize the following: $\tilde{f(x)}$ = $|c_1|^{\top}\vec{x}$+$\vec{x}^{\top}\mathrm{diag}(c_2)\cdot\vec{x}$. The problem turns out to be convex. However, $\tilde{f(x)}$ is the upper bound of the original problem $f(x)$. How can I use the original $f(x)$ instead of its upper bound in the minimization.
Thanks