How to approach a minmax problem?

Question

Starting with a certain geometric problem, I have reached this function:

$$R(s,t,u,v)=\max(s-u,s+u,t-v,t+v,sX+tY+u, tX-sY+v)$$

where $X\geq0$ and $Y\geq0$ are parameters.

I have to find the minimum possible value of $R$ given the paramters, i.e.:

$$\min_{s,t,u,v}R(s,t,u,v) $$

with the given constraints:

$$s\geq 0$$ $$t\geq 0$$ $$s^2+t^2=1$$

How do I approach this problem? How do I approach such min-max problems in general?

EDIT: While I still don't have a general solution, I have made some progress with this particular problem in a particular case: the case in which $u\geq 0$ and $v\geq 0$.

In this case, we can drop the terms $s-u$ and $t-v$ from $R$. Additionally, because $u$ and $v$ are unrelated, we can write $R$ as follows:

$$R(s,t,u,v)=\max( u+\max(s,sX+tY), v+max(t,tX-sY))$$

Now it is clear that, regardless of $s$ and $t$, in order to minimize $R$ we must take $u=v=0$. This gives:

$$R(s,t)=\max(s,sX+tY, t,tX-sY)$$

I.e. the function to minimize is now a function of only 2 variables.

My questions now are:

• Is this simplification process valid?
• Does it generalize to other similar problems, or is it only a lucky coincidence that works only in this particular problem?

Answer 1

Calling it a min-max problems sends the wrong signal, as that typically is reserved for a different type of problems. You have a standard optimization problem, albeit with a non-smooth objective. It can trivially be cast as a problem without any max-operator though by introducing a new variable to upper-bound the element in the max

$$\min_{z,s,t,u,v} z$$ $$z\geq s-u,z\geq s+u,z\geq t-v,z\geq t+v,z\geq sX+tY+u, z\geq tX-sY+v$$ $$s\geq 0,t\geq 0,s^2+t^2=1$$

Your real problem though is the fact that you have a nonconvex constraint on $s$ and $t$.

What is the underlying geometric problem?