There are situations (in optimization problems) where problems are formulated as
$$ f(E_1,E_2 ; \lambda) = E_1 + \lambda E_2 $$
where $E_1,E_2$ are some energy functions, and $\lambda$ is a real value that "weights" the contributions of such energies. I wonder if there's any relationship with a formula of the form
$$ g(E_1,E_2;\lambda) = (1-\lambda)E_1+\lambda E_2 $$
is there maybe some reparametrization that would allow to switch from one form to the other?
Thank you
In general, the optimal solution (but not the objective value) of $\min_x f(x)$ and $\min_x cf(x)$ are the same for $c > 0$. Therefore, $$f(E_1,E_2 ; \lambda_f) = E_1 + \lambda_f E_2$$ and $$h(E_1,E_2 ; \lambda_f) = \frac{1}{1+\lambda_f} E_1 + \frac{\lambda_f}{1+\lambda_f} E_2$$ have the same solution (take $c=1/(1+\lambda_f)>0$). You can see $h$ as $g$ where the $\lambda_g$ in $g$ equals $\lambda_f / (1+\lambda_f)$.