Let $f(x) = \text{argmax} (x^Ty + \dfrac{1}{c}e^{x^Ty})$, what is $\lim f(x)$ as $c \to \infty$

Question

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!

Answer 1

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.