$\|x\|_1$ is a convex function as it's a norm, thus $-\|x\|_1$ is a concave function. The toy optimization problem as follows:
$$\min-\|x\|_1, \forall x \in R^n$$ is not a convex optimization problem as the objective function is concave. However if I try to reformulate it as a linear programming, I get: $$\min_{x,z \in R^n} \{-1^Tz \mid -z \le x \le z \}$$
It seems that I get somehow a linear programming which is a convex problem with an affine objective function. So this is contradictory to the previous statement. There must be something I forget or misunderstand. Could you point it out to me ?