Finding global maximum on non-compact sets using compactification

[SweatingBear]

Been a while since I stopped by here...

There's one thing about optimization on non-compact sets that's been bugging me for quite a while and I'd love to hear how you perceive things.

Say we are optimizing a partially differentiable (and thus continuous) function $f:\mathbb{R^2} \to \mathbb{R}$ over entire $\mathbb{R}^2$. Suppose $f(x,y) \geqslant 0$ for all $(x,y) \in \mathbb{R}^2$ and say at $(1,0)$ we have one stationary point where $f(1,0) = 4$. Furthermore, let's assume that $f(x,y) \to 0$ when $r = \sqrt{x^2 + y^2} \to \infty$.

So we suspect $4$ is the global maximum of the function but how to be certain? Well by definition we have for the previously stated limit

$$\forall \epsilon, \exists \delta : r > \delta \implies |f(x,y)| < \epsilon $$

Especially for $\epsilon = 4$ we can say that there exists some $\delta_0$ such that

$$r > \delta_0 \implies |f(x,y)| < 4 $$

Assume now that $r_0 > \delta_0$ and let us study our function on the set $M = \{ (x,y) : x^2 + y^2 \leqslant r_0^2\}$. Since $M$ is compact and $f$ continuous on it, by extreme value theorem a maximum value must exist on it. Our epsilon-delta-statement tells us that on the boundary of $M$ and outside of it, $f(x,y)$ never equals $4$. So the only possible case left is that $f(x,y) = 4$ on an interior point of $M$

But how do we from here take the leap and argue that $f(x,y) = 4$ is necessarily a global maximum? I have trouble fully understanding why it really is necessary to restrict the domain into a compact region with some specific and fixed $r_0$; how is that really useful, I feel like it does not necessarily provide any further information? Does not the epsilon argument suffice by itself?

 

[ThePerfectHacker]

It is not true.

Keep things simpler by working in $\mathbb{R}$. Find an example of a diff $f:\mathbb{R}\to \mathbb{R}$ where $0$ is a critical point for $f$, and where $f\geq 0$, and where $f(x)\to 0$ as $|x|\to \infty$ but $0$ is not maximum point. Perhaps, you can visualize this function?

The following however is true.

Theorem: Let $f:\mathbb{R}\to \mathbb{R}$ be such that:
(i) $f$ is of class $\mathcal{C}^2$
(ii) $f(x)\to 0$ as $|x|\to \infty$
(iii) $0$ is the only critical point
(iv) $f''(0) < 0$
Then $0$ is a global maximum point for $f$.

Proof: By (iii) and (iv) $0$ is a local maximum for $f$. Let $a = f(0)$. Assume for the time being $a > 0$.

By (ii) there is an $r>0$ such that if $|x|\geq r$ then $|f(x)| < a$. Define $K$ to be the interval $[-r,r]$. On $K$ the function $f$ assumes a maximum value. The maximum value cannot be at the endpoints as $f(\pm r) < a$ and $f(0) = a$. By (iii) there is no critical point in $(-r,r)$ other than $0$ thus we conclude that $0$ is maximum point for $f$. Since $f(x)<a$ for all $|x|\geq r$ it means that $f(0) \geq f(x)$ for all $x\not \in K$ and for all $x\in K$ so that $0$ is a global maximum point for $f$.

I am too lazy to think about $a\leq 0$. Maybe you can continue from there?