I want to maximize $$F(w):=\sum_{i\in I}\sum_{j\in I}\int\lambda({\rm d}x)\int\lambda({\rm d}y)\left(w_i(x)p(x)q_j(y)\wedge w_j(y)p(y)q_i(x)\right)\sigma_{ij}(x,y)|g(x)-g(y)|^2$$ subject to $$\sum_{i\in I}w_i=1,$$ where
- $(E,\mathcal E,\lambda)$ is a measure space
- $f\in\mathcal L^2(\lambda)$
- $I$ is a finite nonempty set
- $w_i:E\to[0,1]$ is $\mathcal E$-measurable for $i\in I$
- $p,q_i$ are probability densities on $(E,\mathcal E,\lambda)$ with $\{p=0\}\subseteq\{f=0\}$, $$\{pf\ne0\}\subseteq\left\{\sum_{i\in I}w_i=1\right\}$$ and $\{q_i=0\}\subseteq\{w_ip=0\}$ for $i\in I$
- $\mu:=p\lambda$
- $(E',\mathcal E',\lambda')$ is a measure space
- $\varphi:E'\to E$ is bijective and $(\mathcal E',\mathcal E)$-measurable with $$\lambda'\circ\varphi_i^{-1}=q_i\lambda$$ for $i\in I$
- $\zeta$ denotes the counting measure on $(I,2^I)$
- $\sigma':(I\times E')^2\to[0,\infty)$ is symmetric and $(2^I\otimes\mathcal E')^{\otimes2}$-measurable with $$\int\sigma'((i,x'),\;\cdot\;)\:{\rm d}(\zeta\otimes\lambda')=1\;\;\;\text{for all }(i,x')\in I\times E'$$ and $$\sigma_{ij}(x,y):=\sigma'((i,\varphi_i^{-1}(x)),(j,\varphi_j^{-1}(y))\;\;\;\text{for all }(i,x),(j,y)\in I\times E$$
- $$\tau_{ij}(x,y):=w_i(x)q_j(y)\sigma_{ij}(x,y)\;\;\;\text{for }(i,x),(j,y)\in I\times E$$
- $$\alpha_{ij}(x,y):=\left.\begin{cases}\displaystyle1\wedge\frac{w_j(y)p(y)q_i(x)}{w_i(x)p(x)q_j(y)}&\text{, if }w_i(x)p(x)q_j(y)>0\\1&\text{, otherwise}\end{cases}\right\}\;\;\;\text{for }(i,x),(j,y)\in I\times E$$
- $$k_{ij}:=\alpha_{ij}\tau_{ij}\;\;\;\text{for }i,j\in I$$ and $$k:=\sum_{i\in I}\sum_{j\in I}k_{ij}$$
- $$r(x):=1-\int\lambda({\rm d}y)k(x,y)\;\;\;\text{for }x\in E$$ and $$\kappa(x,B):=\int_B\lambda({\rm d}y)k(x,y)+r(x)1_B(x)\;\;\;\text{for }(x,B)\in E\times\mathcal E$$
- $g:=\sum_{n=0}^\infty\kappa^nf$ exists in $L^2(\mu)$, where $$\kappa f:=\int\kappa(\;\cdot\;,{\rm d}y)f(y)$$ and $\kappa^n$ denotes the $n$-fold composition
This seems to be a hard optimization problem. Can we use the concrete shape of $g$ to simplify the expression for $F$? In any case, how can we discretize this problem? The space $(E',\mathcal E',\lambda')$ can be assumed to be $\left([0,1)^{\mathbb N_0},\mathcal B([0,1))^{\otimes\mathbb N_0},\mathcal U_{[0,\:1)}^{\otimes\mathbb N_0}\right)$, where $\mathcal U_{[0,\:1)}$ denotes the uniform distribution on $[0,1)$.