This is a desmos plot of the problem I am exploring: https://www.desmos.com/calculator/0rxekqcj26
My first question is how to find the subdifferential of $f(x,y) = \max\{|x|, y + 4\}$. From my plotting, I see that it is just this function $h(x) = |x + 2| + 2$, but I don't think I can just take a subdifferential of this single variable function, which would give me something like $$ \{-1\} \quad x < -2\\ [-1, 1] \quad x = -2\\ \{1\} \quad x > -2 $$ From here, not sure where to go.
Next, we have constraints $g_1(x,y) = (x-1)^2 + (y-1)^2 - 1 \leq 0$ and $g_2(x,y) = |x| + y - b \leq 0$. Of the two constraints, one of them has an arbitrary parameter $b>0$. I'm not sure how to think about a solution that depends on this $b$.