$$f(w,b)=min_{x\ge y\ge w}\{ c_ry-c_rw+\alpha c_ux-\alpha c_uy+\alpha g(x,b)+(1-\alpha)g(y,b)\}$$ where $0\le\alpha\le1$, $g(z,b)$ is complex and there is $c_u$ inside of $g(z,b)$.
What we know: 1. $g(z,b)$ is submodular in $(z,c_u)$ whereas $g(z,b)+c_uz$ is supermodular in $(z,c_u)$. 2. $g(z,b)$ is submodular in $(z,b)$ and supermodular in $(b,c_u)$.
What we try to verify:$f(w,b))$ is supermodular in $(b,c_u)$.(Thank you so much!)