Let $(E,\mathcal E,\lambda),(E',\mathcal E',\lambda')$ be measure spaces, $I$ be a finite nonempty set, $p,q_i$ be probability densities on $(E,\mathcal E,\lambda)$, $\varphi_i:E'\to E$ be bijective and $(\mathcal E',\mathcal E)$ measurable with $\lambda'\circ\varphi_i^{-1}=q_i\lambda$, $w_i:E\to[0,1]$ be $\mathcal E$-measurable with $\sum_{i\in I}w_i=1$, $q':E'\times E'\to[0,\infty)$ be symmetric with $\int\lambda'({\rm d}y')q'(x',y')=1$ for all $x'\in E'$ and $\mu:=p\lambda$.
I want to choose $(w_i)_{i\in I}$ such that $$\sup_{\substack{g\:\in\:\mathcal L^2(\mu)\\g\:\ge\:0\\\left\|g\right\|_{L^2(\mu)}\:\le\:1}}\int\mu^{\otimes2}({\rm d}(x,y))\sum_{i\in I}\sum_{j\in I}w_i(x)w_j(y)\sigma_{ij}(x,y)|g(y)-g(x)|^2\tag1$$ is maximized, where $$\sigma_{ij}(x,y):=\min\left(\frac{q_j(y)}{p(y)}q'\left(\varphi_i^{-1}(x),\varphi_j^{-1}(y)\right),\frac{q_i(x)}{p(x)}q'\left(\varphi_j^{-1}(y),\varphi_i^{-1}(x)\right)\right)$$ for $i,j\in I$ and $x,y\in E$. How can we solve this problem?
Maybe it's a dumb idea, but I wonder whether we can simply pointwise maximize the integrand in $(1)$, ignoring the constant factor $g\in\mathcal L^2(\mu)$, i.e. fix $(x,y)\in E^2$, let $\sigma_{ij}:=\sigma_{ij}(x,y)$ and maximize $$F(v,w):=\sum_{i\in I}\sum_{j\in I}v_iw_j\sigma_{ij}\;\;\;\text{for }(v,w)\in\mathbb R^I\times\mathbb R^I$$ on $\left\{G=0\right\}$, where $$G(v,w):=\left(\begin{array}\displaystyle\sum_{i\in I}v_i-1\\\sum_{j\in I}w_j-1\end{array}\right)\;\;\;\text{for }(v,w)\in\mathbb R^I\times\mathbb R^I.$$ Sure, this is ignoring the nonnegative constraints, which is an issue (can we fix it?). By the Lagrange multiplier theorem, a local extremum $(v,w)$ of $F$ constrained on $\left\{G=0\right\}$ satisfies ${\rm D}\mathcal L((v,w),\Lambda)=0$, where $\mathcal L((v,w),\Lambda):=F(v,w)-\langle\Lambda,G(v,w)\rangle$. For example, if $I=\{1,2\}$, I obtain \begin{equation}\begin{split}v_1&=\frac{\sigma_{22}-\sigma_{21}}{\sigma_{11}-\sigma_{12}-\sigma_{21}+\sigma_{22}},\\ v_2&=\frac{\sigma_{11}-\sigma_{12}}{\sigma_{11}-\sigma_{12}-\sigma_{21}+\sigma_{22}},\\w_1&=\frac{\sigma_{22}-\sigma_{12}}{\sigma_{11}-\sigma_{12}-\sigma_{21}+\sigma_{22}},\\ w_2&=\frac{\sigma_{11}-\sigma_{21}}{\sigma_{11}-\sigma_{12}-\sigma_{21}+\sigma_{22}}.\end{split}\tag2\end{equation} Somehow surprisingly, we note that the obtained solutions $(w_1(x),w_2(x))=(v_1,v_2)$ and $(w_1(y),w_2(y))=(w_1,w_2)$ are compatible.
Is this approach feasible? If so, can we fix the nonnegativity issue? If not, what can we do else?
