# Confusing max(min(f(x,y)) notation?

• Jyan
In summary: This is a valid question. In order to answer it, we need to know a little bit more about the definition of g(x). Let's take a look at the definition again:g(x) = \min_{y\in B} f(x,y).This means that if we take any x in A, and set y to be whatever value is inside the curly brackets, then g(x) will be the smallest value of f(x,y) that exists. This is always a number, because it takes two arguments and each one of them must be a number. So, when we say that g(x) is the smallest value of f(x
Jyan
I'm working on some MIT OCW for probability theory, and I've come across some confusing notation in the assignmnet. Look at exercise 3 here:

http://ocw.mit.edu/courses/electric...y-fall-2008/assignments/MIT6_436JF08_hw01.pdf

I understand what the max and min functions do on their own (find the max or min value of the function subject to the constraints written under the function), but what does it mean when they are combined like this?

Are they applied one after the other? So that max min f(x,y) would mean to find the minimum function f(x) (whatever that means, there is no given measure of the "size" of a function) then find the maximum value of the function f(x). Or are they somehow applied together?

Perhaps $$min_{y \in B} f(x,y)$$ means to find the function f(x) (set y constant) such that f(x) has the smallest minimum value (the global min)? likewise, $$max_{x\in B} f(x,y)$$ means to find the function f(y) (set x constant) such that f(y) has the largest maximum value (the global max)? I don't think this is right though, since it is easy to imagine a counter example for the statement we are supposed to prove.

Anyone know exactly what this notation means?

Thanks for any help.

Last edited:
##\min_{y \in B}f(x,y)## is a function of ##x##. For a given (fixed) value of ##x##, it is the smallest value of ##f(x,y)##, for any value of ##y \in B##.

##\max_{x \in A} \min_{y \in B} f(x,y)## means ##\max_{x \in A} \left( \min_{y \in B} f(x,y)\right)##.

But, I still don't understand completely. Why are there no quantifiers or anything in the problem statement? I don't see how the definition you have given can lead to a sensible interpretation of the problem. If you have some fixed x for the left side of the ≤ sign, what do you do with it on the right side? It loses it's meaning when you go to the right side of ≤.

For $A,B$ finite and $f:A\times B\to\mathbb R$, we're interested in properties of the number $\alpha = \max_{x\in A}\min_{y\in B}f(x,y)$.
-To start, define the function $g:A\to\mathbb R$ via $g(x) = \min_{y\in B} f(x,y)$. This is a well defined number. Indeed, for any given $x\in A$, the right-hand side expression is a precisely defined number.
-Now, $\alpha$ is defined similarly, just as $\alpha=\max_{x\in A}g(x)$.



Now, $\beta = \min_{y\in B}\max_{x\in A}f(x,y)$ is a (possibly different) number too, and so it makes sense to ask whether or not $\alpha\leq\beta$.

I see. Thank you :)

## What is the meaning of max(min(f(x,y)) notation?

The max(min(f(x,y)) notation is a mathematical notation used to represent the maximum value of the minimum values of a function f(x,y). This means that the function is first evaluated at different values of x and y, and then the minimum value from each evaluation is selected. Finally, the maximum value out of these minimum values is determined.

## How is this notation different from other mathematical notations?

This notation is unique because it combines both the maximum and minimum operations in one expression. Other notations may only include one of these operations, making this notation more concise and efficient.

## When is this notation typically used?

This notation is commonly used in optimization problems, where the goal is to find the maximum value of the minimum values of a function. It is also used in game theory, where players aim to maximize their minimum payoff.

## What are some examples of using this notation?

An example of using this notation is in finding the shortest path between two points on a map. The function f(x,y) represents the distance between any two points on the map, and the max(min(f(x,y)) notation would show the maximum distance that needs to be traveled for any path between the two points.

## Are there any limitations to using this notation?

One limitation of this notation is that it can only be applied to functions with two variables. It also does not consider other factors such as time or cost, so it may not be suitable for all optimization problems.

• Calculus
Replies
4
Views
1K
• Calculus
Replies
9
Views
1K
• Calculus
Replies
11
Views
2K
• Calculus
Replies
3
Views
2K
• Calculus
Replies
3
Views
2K
• Calculus
Replies
1
Views
2K
• Calculus
Replies
18
Views
2K
• Calculus
Replies
1
Views
1K
• Calculus
Replies
32
Views
3K
• Calculus
Replies
3
Views
2K