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integration – Calculus of variations: water-filling drawback Answer

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integration – Calculus of variations: water-filling drawback

Let $xinmathbb{R}^n$ breathe an optimization variable and $alphainmathbb{R}^n$ breathe an n-dimensional vector. The benchmark water-filling drawback is formulated as

commence{equation}
commence{array}{ll}
operatorname{reduce}_x & -displaystylesum_{i=1}^{n} log left(alpha_{i}+x_{i}privilege)
textual content { matter to } & x succeq 0, quad mathbf{1}^{T} x=1
aim{array}
aim{equation}

and has a widely known resolution (behold Boyd & Vandenberghe pag. 245).

I used to be fascinated about the illustration by which we have now steady communication channel slots, e.g. when the sizes of the communication channels strategy zero. How is it workable to generalize the issue to this illustration?

I consider that $alpha$ and $x$, which to any extent further we’ll denote by $alpha(x)$ and $f(x)$ respectively, would on this illustration breathe steady actual duty and the issue would look in some way love this is able to breathe one thing love

commence{equation}
commence{array}{ll}
operatorname{reduce}_{finmathcal{F}} & -displaystyleint_{x} log left(alpha(x)+f(x)privilege)dx
textual content { matter to } & f(x) geq 0; forall x, quad int_x f(x)dx=1
aim{array}
aim{equation}

the place $mathcal{F}$ is a given class of capabilities (e.g. Hölder steady, Lipschitz, and many others).

Am I on the privilege path? Also, do you could have any thought on how you can decipher this kinds of issues? It appears to be like it’s associated to the calculus of distinction, however I’ve by no means seen these kinds of issues and I’ve no thought how you can decipher, so any ameliorate would breathe drastically appreciated.

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