Given the multivariate constrained optimization problem
$$ \max_{\mathbf{w}} f(y(\mathbf{w})) \:\:\:\: s.t. \:\:\: g(y(\mathbf{w})) = \overline{g} $$
Where $\mathbf{w}$ is a vector of control variables, $\overline{g}$ is a constant, and $\mathbf{y}$ is a vector of uncoupled functions of the elements $w_i$ in $\mathbf{w}$. More specifically, $y_i = k_i (w_i / C)^{\alpha_i}$, where $0 < \alpha_i < 1$ and $k_i > 0$ are parameters that are different for each $i$, and $C > 0$ is a global parameter.
Can I just reformulate the problem as
$$ \max_{\mathbf{y}} f(\mathbf{y}) \:\:\:\: s.t. \:\:\: g(\mathbf{y}) = \overline{g} $$
And then take the optimal solution $\mathbf{y}^*$ and deduce the optimal $\mathbf{w}^*$ as
$$ w_i^* = C (y_i^* / k_i)^{1 / \alpha_i} $$