Can we simplify objective function based on the property of optimal solution?

Question

Consider the non-convex optimization problem \begin{equation} \begin{aligned} \max_{x} & \quad f(x)\\ s.t. & \quad 0 \leq x\leq 1 \end{aligned} \tag{1} \end{equation}

where $f(x)$ is non-concave. But $\forall y \in X = \{x| Ax=b\}$ we have $f(y) = g(y)$, where $g(y)$ is concave. We know the optimal solution $x^* \in X$ .

So can we transform the non-convex optimization problem into the following convex problem?

\begin{equation} \begin{aligned} \max_{x} & \quad g(x)\\ s.t. & \quad Ax = b\\ & \quad 0 \leq x\leq 1 \end{aligned} \end{equation}

Answer 1

Yes. The following are equivalent problems:

$$\max_x f(x) \quad \text{subject to} \quad 0 \le x \le 1$$

$$\max_x f(x) \quad \text{subject to} \quad x\in X,\ 0 \le x \le 1$$

$$\max_x f(x) \quad \text{subject to} \quad Ax=b,\ 0 \le x \le 1$$

$$\max_x g(x) \quad \text{subject to} \quad Ax=b,\ 0 \le x \le 1$$