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Let's suppose that I wanted to compute $\left\|f\right\|_{\infty}=\sup_{t \in \mathcal{T}} \left|f(t)\right|$ for a $f$ that may not be easy to optimize. This is the infinity norm of a function, and there's an asymptotic relationship: as $p \to \infty$, $\left\|f\right\|_p \to \left\|f\right\|_{\infty}$ with $\left\|f\right\|_p = \left(\int_{\mathcal{T}}\left|f(t)\right|^p dt\right)^{1/p}$. This relationship suggests we could attempt to solve the optimization problem (in the sense of finding the maximum) via numerical integration, computing the integral for a very large $p$ and taking the $p^{\text{th}}$ root.

How well would this approach to estimating $\left\|f\right\|_{\infty}$ work numerically?

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    $\begingroup$ Depending on the set $\mathcal{T}$ and the regularity of $f$, estimating this integral to desired accuracy may be functionally equivalent to finding the supremum $\endgroup$ Commented Oct 23 at 18:07

2 Answers 2

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It converges very slowly... The example below is for $f(x)=x^2$ in the interval $[0,1]$. The integrals are computed analitically.

$$ \left( \begin{array}{cc} p & \textrm{Error}\\ 1. & 0.666667 \\ 2. & 0.552786 \\ 3. & 0.477242 \\ 4. & 0.42265 \\ 5. & 0.380956 \\ 6. & 0.347857 \\ 7. & 0.320817 \\ 8. & 0.298231 \\ 9. & 0.279032 \\ 10. & 0.262473 \\ 11. & 0.24802 \\ 12. & 0.235276 \\ 13. & 0.22394 \\ 14. & 0.213782 \\ 15. & 0.204618 \\ 16. & 0.196302 \\ 17. & 0.188717 \\ 18. & 0.181766 \\ 19. & 0.175369 \\ 20. & 0.16946 \\ \end{array} \right) $$

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    $\begingroup$ More precisely, $$ \|x^a\|_{L^\infty} - \|x^a\|_{L^p} = 1 - (ap+1)^{-1/p} = \frac{\ln p}{p} + \frac{\ln a}{p} + O(1/p^2) $$ $\endgroup$ Commented Oct 24 at 8:14
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To sort of complement the above answer, consider a very simple discrete problem and set $a = \max\{a_1,\dots a_m\}$ and $a_1<a_2<\dots a_m$, then $$a-\sqrt[n]{\dfrac{a_1^n+\dots + a_m^n}{n}}\sim \dfrac{a\ln m}{n}+\dfrac{a}{n}\left(\dfrac{a_{m-1}}{a}\right)^n + O(n^{-2}),$$ so the error term is linear in $n^{-1}$ which isn't that fast.

I do think you can formalize this for the case of nice integrable functions over a compact interval.

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