Timeline for Maximize a Lebesgue integral subject to an equality constraint
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 27, 2019 at 19:53 | comment | added | 0xbadf00d | I was able to reduce my problem to a smooth optimization problem. It would be great if you could took a look mathoverflow.net/q/344621/91890. | |
| Sep 24, 2019 at 15:28 | comment | added | 0xbadf00d | Give me one last shot: $x\mapsto x^+:=\max(0,x)$ can be smoothly approximated by $x\mapsto p_\alpha(x):=x+\frac1\alpha\ln\left(1+e^{-\alpha x}\right)$, $\alpha>0$. We could use this to approximate $x\mapsto|x|=x^++(-x)^+$ and hence to approximate $(a,b)\mapsto\min(a,b)=\frac12(a+b-|a-b|)$. We could solve the optimization problem with an accordingly modified objective function. If $w^{(\alpha)}$ is the resulting maximizer, how would it be related to the maximizer $w$ of the actual objective function? (If you know it, this might be an answer to one of the other questions.) | |
| Sep 24, 2019 at 6:24 | comment | added | 0xbadf00d | interchange of minimization and integration given in Theorem 14.60 of the book Variational Analysis by Rockafellar and Wets (see also Theorem 2.1 in this paper) might be useful. Do you think it is applicable here? | |
| Sep 23, 2019 at 13:47 | comment | added | 0xbadf00d | Sorry, I'd missed the point that $w^\ast$ is implicitly involved in $(7)$ in the definitions of the $\theta_i$. Do you've got an idea how we might proceed from here? | |
| Sep 23, 2019 at 6:16 | comment | added | 0xbadf00d | Is the equality $(3)$ of the normal cone correct? This is the only part where I wasn't sure. If it is correct, do you see any other option for solving the problem? | |
| Sep 23, 2019 at 5:55 | comment | added | dchatter | It appears that the necessary conditions don't give much information in this case. | |
| Sep 22, 2019 at 18:54 | comment | added | 0xbadf00d | Just worked out the left details (mathoverflow.net/q/342173/91890) and noticed that $w$ disappears when we differentiate the integral function. So, it seems like your approach is of no help to determine the maximizing $w$. Maybe you can take a look if I missed something crucial. | |
| Sep 21, 2019 at 14:53 | comment | added | 0xbadf00d | I've tried to figure out as much as possible in the special case $I=\{1,2\}$ and asked a separate question for how we can proceed: mathoverflow.net/q/342173/91890. Would be great if you could take a look. | |
| Sep 17, 2019 at 19:03 | comment | added | 0xbadf00d | Thanks your your reply. If I understand Clarke's text correctly, the necessary condition is a generalized version of the usual Lagrage multiplier rule in Banach spaces. But as I wrote before, my main problem is the application of the rule. It seems quite complicated to treat the (generalized) derivative of the integrand. If you know how to do it, it would be great if you could at least briefly outline how we need to proceed. | |
| Sep 17, 2019 at 16:16 | comment | added | dchatter | My apologies for the delay. The concepts involved here are the Clarke generalized directional derivatives, limiting cones, and the like, and they're fine-tuned to handle Lipschitz nonsmoothness like the max; these constructions have been treated at length in Clarke's text that I mentioned. | |
| Sep 15, 2019 at 19:15 | comment | added | 0xbadf00d | I know how to calculate the Fréchet/Gâteaux derivative; that's not the problem. My problem is that the integrand is not everywhere differentiable (due to the occurrence of $\min$) and it seems quite complicated to derive a formula we can work with. If you know how to do it, this would be a great addendum to your answer. | |
| Sep 15, 2019 at 19:08 | comment | added | dchatter | Standard manipulations are needed to make the max into a min, that's hardly an issue. You will need derivatives on Banach spaces, sure; this is again standard material in functional analysis textbooks. | |
| Sep 15, 2019 at 17:08 | comment | added | 0xbadf00d | I've taken a quick look at the theorem in the book. Actually, I thought about a Lagrange multipliers approach before, but how exactly would we apply it here (which Banach/Hilbert space do we choose and how do we calculate the derivatives)? (My problems in applying the Lagrange multiplier approach led me to the search for a sharp lower bound for which the Lagrange multiplier approach is easier to apply, but I wasn't able to find a suitable bound yet.) | |
| Sep 15, 2019 at 17:04 | comment | added | 0xbadf00d | You surely mean the $\min$ (not the $\max$) in the objective functional. | |
| Sep 15, 2019 at 16:28 | history | answered | dchatter | CC BY-SA 4.0 |