Keep parameters unspecified until it is clear how to optimize them

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Keep parameters unspecified until it is clear how to optimize them

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[QUICK DESCRIPTION]

In the course of proving some statement, you may want to introduce some parameter (e.g. if you wish to divide into cases when a certain variable $x$ is small or large, one may introduce a parameter $R$ and divide into the cases $|x|<R$ and $|x| \geq R$.)  But you don't yet know what the "best" choice of this new parameter is.  In many cases, what one can do is just leave the parameter unspecified, and continue the argument with this undetermined parameter.  At a much later stage of the argument, it may become clearer what metric one would use to decide what choices of parameter are "good" and what are "bad", at which point one can ''optimize'' in that metric and find the right choice of parameter.

[PREREQUISITES]

[EXAMPLE]

Suppose one wishes to establish the Cauchy-Schwarz inequality

[math] \sum_{n=1}^N a_n b_n \leq \| a \| \|b\|[/math]

for non-negative real numbers $a_1,\ldots,a_N,b_1,\ldots,b_N$, where

[math] \|a\| := (\sum_{n=1}^N |a_n|^2)^{1/2}; \quad \|b\| := (\sum_{n=1}^N |b_n|^2)^{1/2}.[/math]

A first attempt would be to use the arithmetic mean-geometric mean inequality

[math amgm] a_n b_n \leq \frac{1}{2} a_n^2 + \frac{1}{2} b_n^2[/math]

but when one substitutes this in and works everything out, one gets the inequality

[math] \sum_{n=1}^N a_n b_n \leq \frac{1}{2} \| a \|^2 + \frac{1}{2} \|b\|^2.[/math]

which is inferior to Cauchy-Schwarz (by another application of the AM-GM inequality!).

However, one can be cleverer about this by multiplying $a_n$ by an unspecified parameter $\lambda > 0$ and dividing $b_n$ by the same parameter, which when inserted into [eqref amgm] gives the more general inequality

[math amgm2] a_n b_n \leq \frac{\lambda^2}{2} a_n^2 + \frac{1}{2\lambda^2} b_n^2.[/math]

It is not yet clear what value of $\lambda$ is best (though we already know that $\lambda=1$ does not always work).  But one can forge on regardless.  Summing [eqref amgm2] in $n$, one obtains

[math] \sum_{n=1}^N a_n b_n \leq \frac{\lambda^2}{2} \|a\|^2 + \frac{1}{2\lambda^2} \|b\|^2.[/math]

And now it is clear what to do to optimize in $\lambda$: one should choose $\lambda$ so that the right-hand side is as small as possible.  A little calculus (see also the heuristic "[[To optimize a sum, try making the terms roughly equal in size]]") then shows that the optimal choice of $\lambda$ is $\|b\|^{1/2}/\|a\|^{1/2}$, at least when $\|a\|$ and $\|b\|$ are both non-zero (but the case when $\|a\|=0$ or $\|b\|=0$ can be easily handled separately).  Inserting this choice of $\lambda$ gives the desired inequality.

[GENERAL DISCUSSION]

This is a special case of "[[If you don't know how to make a decision, then don't make it]]".

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Keep parameters unspecified until it is clear how to optimize them

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