Describe the solution of $$\min_w \max \{ c_1\|A_1w-b_1\|_2^2, c_2\|A_2w-b_2\|_2^2\}$$ for some positive constants $c_1, c_2$.
Is there a keyword I can search or a reference I can look into for this problem? Below has been my attempt so far:
The optimization problem is convex and the solution is either the min of either equation or within their intersection i.e. when $c_1\|A_1w-b_1\|_2^2= c_2\|A_2w-b_2\|_2^2$. That condition turns out to be $w^T(c_1A_1^TA_1-c_2A_2^TA_2)w+ 2[c_2b_2^TA_2 - c_1b_1^TA_1]w+\|c_1b_1\|_2^2 - \|c_2b_2\|_2^2 = 0$. The problem is not strongly convex, so there might be infinitely many solutions. We can also consider the "regularized" version of the problem so that the problem becomes strongly convex and then there is a unique optimal solution.
I can see that the solution must be either "inside" (I want to say convex hull but not sure if that's correct) or exactly in the intersection $I=\{w \mid c_1\|A_1w-b_1\|_2^2= c_2\|A_2w-b_2\|_2^2\}$ -- since the objective is convex -- and that when the solution is not on the boundary of that set, it is the min of either quadratic. This intersection set $I$ is not a convex set nor even it is the boundary of its convex hull (when $c_1A_1^TA_1-c_2A_2^TA_2$ is neither PSD nor NSD which seems possible) so that makes things more complicated.
My intuition is that when $c_1A_1^TA_1-c_2A_2^TA_2$ is PSD (or NSD) i.e. the intersection set $I$ is the boundary of its convex hull (is there a name for such a set?), then the problem is a constraint optimization problem with the equality constraint to check for if the solution is in $I$ or not. If not then it is a simple casework to figure which quadratic has the min.
Am I on the right track? Any pointer would be greatly appreciated!