I am presented with this function
$f(x) = \text{argmax}_{y \in \mathbb{R}^n} (x^Ty + \dfrac{1}{c}e^{x^Ty})$, where $c$ is a constant
My question is what is $\lim_{c \to \infty} f(x)$?
My main point of confusion is that:
$\lim_{c \to \infty} f(x) =\lim_{c \to \infty} \text{argmax}_{y \in \mathbb{R}^n} (x^Ty + \dfrac{1}{c}e^{x^Ty})$
Under what condition can we move the limit INSIDE of the argument? $ \text{argmax}_{y \in \mathbb{R}^n} (x^Ty + \lim_{c \to \infty} \dfrac{1}{c}e^{x^Ty})$
If I cannot move the limit inside the argument, then I have no way of knowing how to evaluate this limit. Help!
You can't generally move the limit inside. It could be that the second term always controls what the max is, regardless of $c.$ As a counterexample consider $\mathrm{argmax}(1+(x-2)^2/c).$ or $\mathrm{argmax}(-x^2+e^x/c).$ That being said, I can't think of any examples $\mathrm{argmax}(f(x)+g(x)/c)$ where both $f$ has a unique maximum and $g$ has a finite maximum where the limit doesn't pass.
However, here you have $x^Ty + e^{x^Ty}/c$ which is an increasing function of $x^Ty$ for any $c>0.$ Thus it will always be maximized when $x^Ty$ is maximized. So $\mathrm{argmax}(x^Ty + \exp(x^Ty))$ is given by $\mathrm{argmax}(x^Ty)$, but this particular logic doesn't have anything to do with bringing the limit inside.