Finding a solutions for an equation

integration – Continuous water-filling drawback Answer

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integration – Continuous 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 answer (behold Boyd & Vandenberghe pag. 245).

I used to be eager about the illustration during which we’ve 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 $x$ would on this illustration breathe steady actual duty and the issue would look in some way love this might 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 features (e.g. Hölder steady, Lipschitz, and so forth).

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

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