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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
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
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)?
Jul
16
comment Dominated convergence theorem for complex-valued functions?
@ConradoCosta You see, I am having trouble locating that term that you wrote down -- it's not in my calculations anywhere. Perhaps you are confusing $a_-^2$ with $a_S^2$ and $a_+^2$ with $a_L^2$? In that case, note the negative sign in front of $\frac{it}{2a_L}$ in the term on line 2 corresponding to $a_-^2$... You can see that most of the terms in line 3 either combine or cancel out when you do the arithmetic...
Jul
16
comment Dominated convergence theorem for complex-valued functions?
@ConradoCosta I am not sure where in my derivation you found that expression. In the first block of formulas of "specific example" the second equality is from Taylor series expansion of $e^x$; the third equality just plugs in the definitions of $a_-^2$ and $a_+^2$, and expands the squares; the fourth equality is just careful arithmetic -- a lot of terms cancel out; and final equality just evaluates the limit.
Jul
16
comment Dominated convergence theorem for complex-valued functions?
"Example" refers to the "specific example" I gave in my question (sorry that it's long). Your answer allows me to interchange the limit and the integral at the very end of the "specific example." This shows that the probability density function of the random variable $A$ (which you recover from the characteristic function using Fourier transform) approaches Gaussian in the limit (after dividing by $2\pi$, which I omitted for brevity). Convergence in probability described earlier in "specific example" is central-limit-theorem-like, while convergence of densities is "local". Does this explain?
Jul
16
revised Dominated convergence theorem for complex-valued functions?
Fixed a minor typo in the last line by moving the symbol indicating the real part inside the limit.