In Birge's stochastic optimization book, we have the following formulation where $\zeta$ is a random variable:
$z(x,\zeta) = c^Tx + \min \{q^Ty|Wy = h -Tx, y\geq 0 \} \\ s.t. \ AX = b, x \geq0$
In this model $W,h,T,q$ are all dependent on the scenario $\zeta$. However, when we look for the fixed recourse case, $W$ and $q$ are fixed parameters.
Now the question is in Birge's book, it is said that in the fixed recourse case $z(x,\zeta)$ is jointly convex in $x$ and $\zeta$. The explanation is the following:
The problem can be written as
$z(x,h,T) = c^Tx + Q(x,h,T) + \delta(x | Ax = b, x\geq0 )$
where $\delta(x|X)$ is the of the point $x$ for set $X$. This $z(x,h,T)$ is concluded to be jointly convex in $(x,h,T)$ or $(x,\zeta)$.
I don't see how one can conclude immediately.
- What is the indicator function? How is it defined?
- What is $Q$?
- How can we conclude it is convex?
- Is it different to say "it is convex in both $x,\zeta$" than "it is jointly convex in these two variables"?
The indicator function takes the value $0$ if $Ax=b, x \geq 0$, and the value $\infty$ otherwise.
Just by comparing functions, it must be that $Q(x,h,T) = \min \{q^Ty|Wy = h -Tx, y\geq 0 \}$.
$z$ is convex since it is the sum of convex functions. The first function is linear, the last function is clearly convex, but I struggle with $Q$ which is not jointly convex in $(x,T)$.
Convex in both $x,\zeta$ is ambiguous. For example, is $f(x,\zeta) = x\zeta$ convex in both $x$ and $\zeta$?