Aug
23
comment Calculation of a Frechet derivative
@NormalHuman Thanks for the comment! So, Gateaux derivative at $X$ in the direction $\psi=(y_i), i=1,2,3,\ldots$ is $dF(X,\psi)=\sum_{i=1}^\infty \frac{y_i}{a_i+x_i}$. However, I am not sure how to show that this is continuous, and what continuity here even means. The wikipedia article on Gateaux derivative (en.wikipedia.org/wiki/G%C3%A2teaux_derivative) is rather confusing. Could you please clarify?
Aug
23
revised Displaying "Supplementary References" instead of a line as a title of a REVTeX bibliography
solved part of the problem, sort-of
Aug
23
asked Displaying "Supplementary References" instead of a line as a title of a REVTeX bibliography
Aug
20
awarded Socratic
Aug
19
asked Calculation of a Frechet derivative
Aug
19
revised Optimizing over an infinite set of variables
typo
Aug
18
revised Optimizing over an infinite set of variables
specified that the sequence (x_i) lives in l^2
Aug
18
comment Optimizing over an infinite set of variables
@user251257 Ahh. Thinking about this deeper, $(x_i)_{i\in\mathbb{N}}$ is in the $\ell^2$ space. I'll update my question.
Aug
18
comment Optimizing over an infinite set of variables
@user251257 Both $f(x)$ and $g(x)$ are convex, which makes my problem convex. They are also differentiable, which, I believe, makes sums Frechet-differentiable (though I am not sure about that). I've updated the question once again. Seems like this isn't as elementary as thought it would be...
Aug
18
revised Optimizing over an infinite set of variables
specify that the problem is convex
Aug
18
comment Optimizing over an infinite set of variables
@user251257 Could you please elaborate on the "if you have frechet diifferentiability, then Lagrange multiplier works basically like in $\mathbb{R}^n$? I think you might be talking about the calculus of variations, which I don't know at all (I would like to start learning it though). Perhaps you could point out to me what needs to be frechet differentiable for me to use Lagrange multipliers?
Aug
18
revised Optimizing over an infinite set of variables
deleted 8 characters in body
Aug
18
comment Optimizing over an infinite set of variables
@user251257 $x_i$'s are real numbers ($x_i^*$'s come out positive, which makes sense withing the greater scope of the problem). I can show that $\lambda\geq0$ exists that satisfies the constraint (I updated the question).
Aug
18
revised Optimizing over an infinite set of variables
typo
Aug
18
comment Optimizing over an infinite set of variables
@user251257 I obtain a functional form for $x_i^*$ (it turns out to be a function of some constant parameters that define $f(x)$ and $g(x)$) and substitute into the objective function and the constraint. I obtain convergence by standard tests -- integral test, with the integral converging by a limit test. (note, there were two typos: minor one in the convergence statement for the objective function, and a bigger one -- the constraint is an inequality).
Aug
18
revised Optimizing over an infinite set of variables
fixed a typo
Aug
18
asked Optimizing over an infinite set of variables
Jul
29
awarded Popular Question
Jul
21
awarded Nice Question
Jul
16
comment Dominated convergence theorem for complex-valued functions?
@ConradoCosta Looks like you missed the negative sign: $\frac{it}{2a_L}\neq -\frac{it}{2a_L}$, so your $A$ doesn't get multiplied by your $C$. It gets multiplied by $\frac{it}{2a_L}-\frac{t^2}{8a_L^2}+O(a_L^{-3})$, which does not equal your $C$. Does that make things clear(er)?
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