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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?

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    Yes, since your $\inf$ still contains the constraint $g(x) = 0$. Is this intended? – 2017-01-26
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    You need $\lambda \in \mathbb{R}$ instead of nonnegative. And your last operation changes the objective value, whereas the first does not. – 2017-01-26
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    If $f$ and $g$ are non-negative then this should not be an issue right? – 2017-01-26
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    Not an issue for $\lambda \geq 0$ because then $g(x)=0$ is equivalent to $g(x) \leq 0$. However, the other comments (including gerw's) are still valid. – 2017-01-27

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