Suppose I want to solve the constrained optimization problem
$$
\inf_{\{x \in \mathbb{R}^d: g(x)=0\}} f(x),
$$
where $f$ is convex and $g$ is lsc.
I know I can rewrite it as a minimax problem
$$
\inf_{\{x \in \mathbb{R}^d: g(x)=0\}} \sup_{\lambda \in [0,\infty)}f(x) + \lambda g(x).
$$
Can I rewrite this as $$ \inf_{\{x \in \mathbb{R}^d: g(x)=0\}} \sup_{\lambda \in (0,1])} (1-\lambda) f(x) + \lambda g(x)? $$ If not, under what conditions would this be possible?