Let $(E,\mathcal E,\mu)$ be a probability space, $I$ be a finite nonempty set, $\gamma:(E\times I)^2\to[0,\infty)$ be measurable, $$F_1(g,w):=\sum_{i\in I}\int\mu({\rm d}x)w_i(x)g(x)\sum_{j\in I}\int\mu({\rm d}y)w_j(y)\gamma((x,i),(y,j))(g(y)-g(x))$$ and $$F_2(g,w):=\sum_{i\in I}\int\mu({\rm d}x)w_i(x)\left|\sum_{j\in I}\int\mu({\rm d}y)w_j(y)\gamma((x,i),(y,j))(g(y)-g(x))\right|^2$$ for $g\in\mathcal L^2(\mu)$ and $w\in\mathcal L^2(\mu)^I$.
I need to minimize $$F(w):=\sup_{\substack{g\:\in\:\mathcal L^2(\mu)\\g\:\ge\:0\\\int g\:{\rm d}\mu\:=\:0}}\left[\int|g|^2\:{\rm d}\mu+2F_1(g,w)+F_2(g,w)\right]\;\;\;\text{for }w\in L^2(\mu)^I$$ over $$C:=\left\{w\in L^2(\mu)^I:\sum_{i\in I}w_i=1\;\mu\text{-almost surely}\right\}.$$ How can we do this? Is a multiplier rule applicable? We may note that $C$ is closed, convex and has empty interior.
EDIT
Fréchet derivatives of $F_1(g,\;\cdot\;)$ and $F_2(g,\;\cdot\;)$ for a fixed $g\in\mathcal L^2(\mu)$:
$${\rm D}_2F_1(g,w)h=\sum_{i\in I}\int\mu({\rm d}x)g(x)\sum_{j\in I}\int\mu({\rm d}y)\gamma((x,i),(y,j))(g(y)-g(x))(w_i(x)h_j(y)+w_j(y)h_i(x))$$ and \begin{equation}\begin{split}&{\rm D}_2F_2(g,w)h=\sum_{i\in I}\int\mu({\rm d}x)\left[h_i(x)\left|\sum_{j\in I}\int\mu({\rm d}y)w_j(y)\gamma((x,i),(y,j))(g(y)-g(x))\right|^2\right.\\&\;\;\;\;\left.+2w_i(x)\left(\sum_{j\in I}\int\mu({\rm d}y)w_j(y)\gamma((x,i),(y,j))(g(y)-g(x))\right)\sum_{j\in I}\int\mu({\rm d}y)\gamma((x,i),(y,j)((y)-g(x))h(y)\right]\end{split}\end{equation} for all $h,w\in\mathcal L^2(\mu)^I$.
