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I want to minimize $$F(w):=\sum_{i\in I}\sum_{j\in I}\int\lambda({\rm d}x)w_i(x)p(x)\int_{\left\{\:pq_j\:>\:0\:\right\}}\lambda({\rm d}y)\frac{\left|w_j(y)p(y)\right|^2}{q_j(y)\sigma_{ij}(x,y)}\left|\frac{f(y)}{p(y)}-\lambda f\right|^2\;\;\;\text{for }w\in L^2(\mu)^I$$ over the set $$C:=\left\{w\in L^2(\mu)^I:\sum_{i\in I}w_i=1\right\}.$$ We easily see that the Fréchet derivative of $F$ at $w$ is given by $${\rm D}F(w)h=\sum_{i\in I}\sum_{j\in I}\int\lambda({\rm d}x)p(x)\int_{\left\{\:pq_j\:>\:0\:\right\}}\lambda({\rm d}y)\frac{\left|p(y)\right|^2}{q_j(y)\sigma_{ij}(x,y)}\left|\frac{f(y)}{p(y)}-\lambda f\right|^2\left(\left|w_j(y)\right|^2h_i(x)+2w_i(x)w_j(y)h_j(y)\right)\tag1$$ for all $w,h\in L^2(\mu)^I$. Using the identificaion $\mathfrak L(L^2(\mu)^I,\mathbb R)\cong L^2(\mu)^I$ we may write \begin{equation}\begin{split}&\left({\rm D}F(w)\right)_i(x)\\&\;\;\;\;=\sum_{j\in I}\int_{\left\{\:pq_j\:>\:0\:\right\}}\lambda({\rm d}y)\frac{\left|w_j(y)p(y)\right|^2}{q_j(y)\sigma_{ij}(x,y)}\left|\frac{f(y)}{p(y)}-\lambda f\right|^2\\&\;\;\;\;\;\;\;\;+2\cdot 1_{\left\{\:pq_i\:>\:0\:\right\}}(x)\frac{w_i(x)p(x)}{q_ix)}\left|\frac{f(x)}{p(x)}-\lambda f\right|^2\sum_{j\in I}\int\lambda({\rm d}y)\frac{w_j(y)p(y)}{\sigma_{ij}(x,y)}\end{split}\tag2\end{equation} for all $i\in I$, $x\in E$ and $w\in L^2(\mu)^I$.

How can we apply the Lagrange multiplier rule and determine the minimizer $w$?

My first problem is that I don't know how I need to incorporate the equality constraint given by the definition of $C$.


Definitions:

  • $I$ is a finite set;
  • $p,q_i$ are probability densities on a measure space $(E,\mathcal E,\lambda)$;
  • $\mu:=p\lambda$;
  • $f\in\mathcal L^2(\lambda)$ with $\{p=0\}\subseteq\{f=0\}$;
  • $\lambda f:=\int f\:{\rm d}\lambda$;
  • $w_i:E\to\mathbb R$ is $\mathcal E$-measurable with $\{q_i=0\}\subseteq\{w_ip=0\}$ for $i\in I$ with $\{pf\ne0\}\subseteq\left\{\sum_{i\in I}w_i=1\right\}$;
  • $\sigma_{ij}:E^2\to\mathbb R$ is $\mathcal E^{\otimes2}$-measurable with $\sigma_{ij}(x,y)=\sigma_{ji}(y,x)$ for all $(i,x),(j,y)\in I\times E$ and $\sum_{j\in I}\int\lambda({\rm d}y)w_i(x)q_j(y)\sigma_{ij}(x,y)=1$ for all $(i,x)\in I\times E$.

Remark: I'm actually searching for $\mathcal E$-measurable $w_i:E\to\mathbb R$ with $\{q_i=0\}\subseteq\{w_ip=0\}$ for all $i\in I$, $\{pf\ne0\}\subseteq\left\{\sum_{i\in I}w_i=1\right\}$ and minimizing $F(w)$. I guess the description above is the best way to formulate this as an optimization problem, but please let me know if you think that I should search for $w$ in a different Banach space or use a different set $C$ (maybe incorporating some of the other requirements mentioned before).

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If $w$ is a minimizer then it holds $$ F'(w)h = 0 $$ for all $h$ such that $\sum_{i\in I}h_i=0$ (i.e. for all $h=u-w$ with $u\in C$). To see this, fix $h$ with $\sum_{i\in I}h_i=0$. Then $w+th$ is in $C$ for all $t$, and $F(w+th)-F(w)\ge0$. Dividing by $t$ and passing to the limits $\searrow0$ and $t\nearrow0$ gives the claim.

Define the mapping $T: (L^2)^I \to L^2$ by $$ Th = \sum_{i\in I}h_i. $$ Now the optimality condition above implies $F'(w) \in N(T)^\perp$ (where $N$ is null-space). By the closed range theorem, $F'(w) \in R(T^*)$, i.e., there is $z\in L^2$ such that $F'(w) = T^*z$ or equivalently $$ F'(w) h = \sum_i (h_i ,z), $$ or $$ F'(w)_i = z, $$ where $F'(w)_i$ is such that $ F'(w)h = \sum_{i\in I} F'(w)_ih_i$.

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  • $\begingroup$ (a) Am I missing something or doesn't ${\rm D}F(w)h=0$ hold for all $h\in L^2(\mu)^I$ (not only those satisfying $\sum_{i\in I}h_i=0$)? (b) The rest is clear to me except what exactly "$F'(w)_i=z$" means. I guess you're identifying $\mathfrak L(L^2(\mu)^I,\mathbb R)\cong L^2(\mu)^I$, but what is ${\rm D}F(w)$ under this identification precisely? (c) How can we proceed to find a closed form expression of a minimizer? $\endgroup$ – 0xbadf00d Oct 25 at 17:36
  • $\begingroup$ Regarding (b): Added the identification in the question (Hopefully didn't make any mistake. $\endgroup$ – 0xbadf00d Oct 25 at 18:11
  • $\begingroup$ (a) no, see edit, (b) I use this identification, also see edit, (c) no idea. Is the functional convex? It seems to be cubic in $w$. $\endgroup$ – daw Oct 25 at 20:12
  • $\begingroup$ (b) So, you use the same identification as I am, right? (c) You mean the functional $F$? Not sure. What can we infer from $({\rm D}F(w))_i=z$ for all $i\in I$? For example, what could we do in the case $I=\left\{1\right\}$ where we trivially know that the solution is $w_1=1$. $\endgroup$ – 0xbadf00d Oct 26 at 8:14
  • $\begingroup$ (d) Couldn't we minimize the integrand in the definition of $F(w)$ pointwise? This would yield this problem: math.stackexchange.com/q/3409501/47771. $\endgroup$ – 0xbadf00d Oct 27 at 7:13

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