I have seen various definitions of quantum conditional min-entropy, which I believe are equivalent.
$$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B \right)$$ where in general $$D_{\max}(\rho \parallel \sigma) = \inf \{ \lambda : \rho \leq 2^\lambda \sigma \}$$
$$H_{\min}(A|B) = \max\limits_{\sigma_B} \sup \{ \lambda: \rho_{AB} \leq 2 ^{-\lambda} I_A \otimes \sigma_B \}$$
Why are the two definitions equivalent?
Observe that $$ H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \| I_A \otimes \sigma_B \right) = \sup\limits_{\sigma_B} [-D_{\max} \left( \rho_{AB} \| I_A \otimes \sigma_B \right)], $$ and $$-D_{\rm max}(\rho\|I\otimes \sigma)=-\inf \{\lambda: \,\, \rho\le 2^\lambda (I\otimes \sigma)\} \\ = \sup\{-\lambda: \,\, \rho\le 2^\lambda (I\otimes\sigma)\} \\ = \sup\{\lambda: \,\, \rho\le 2^{-\lambda} (I\otimes\sigma)\}.$$
It might help to take a slightly more abstract perspective here. What we're trying to compute is $-\inf_x f(x)$ where $f(x)\equiv \inf\{y\in\mathbb{R}:\,\, y\in g(x)\}$ for some function $f$, and some set $g(x)\subseteq\mathbb{R}$ that depends on $x$. And then $$-\inf_x f(x) = \sup_x[-f(x)] = \sup_x [-\inf\{y\in\mathbb{R}:\,\, y\in g(x)\}] \\ = \sup_x \sup\{-y: \,\, y\in\mathbb{R},\,\, y\in g(x)\} \\ = \sup_x \sup\{z\in\mathbb{R}: \,\, -z\in g(x)\}.$$ In words, we first minimise over the possible values of a function $f$ (restricting the possible inputs $x$ in a specific subset, and taking the opposite of the result), where $f(x)$ is defined as the smallest element of a subset of real numbers that depends on $x$. You can then just replace $\sup\to\max$ and $\inf\to\min$ whenever you know that a min/max exists.
