How to make sense of the simple optimization problem $\min_{x,y} x^2+y^2$?

Question

Solve

$$\begin{array}{ll} \text{minimize} & x^2 + y^2\\ \text{subject to} & x \neq y\\ & x + y = 2\end{array}$$

The solution does not exist; however, in practice, we can say that the solution is something like $x=1+\epsilon$ and $y=1-\epsilon$.

Of course, this explanation is not mathematically rigorous. SO my question is, how to rigorously write out this question and answer? Are there any related books that I can read?

Answer 1

As you observe, the minimum does not exist, because no matter which $(x,y)$ you pick, there is always a better one (by moving closer to the line $y=x$). If you want to speak of minimum "in the limit," you can speak of the greatest lower bound of $x^2+y^2$, which mathematicians sometimes call infimum, written $\inf$ instead of $\min$.

Answer 2

We have that by the constraints

$$g(x)=x^2+y^2=x^2+(2-x)^2=2x^2-4x+4 \implies g'(x)=4x-4=0 \implies x=1$$

therefore $g(x)\ge 2 \implies f(x,y)\ge 2$ and equality holds if and only if $x=y=1$.

In this case we say that $f(x,y)$ has not a minimum and that its infimum $\inf(f)=2$.

Answer 3

One of the defining properties of the reals is the LUB/GLB property. Given any set that is bounded above, there exists a supremum, a least upper bound, and a set bounded below has an infimum, a greatest lower bound.

So, many sets, such as yours, doesn't have a minimum value, but has an infimum which is $2$. Only compact sets in the reals are guaranteed to have a maximum and minimum value.