I am reading the book "Metaheuristics From Design to Implementation" written by El-Ghazali Talbi and on page 91, in the "Single-solution based metaheuristics" section, he says that "a problem may have many global optimal solutions." The picture pertinent with the phrase in the book is depicted below:
As far as I know, in a 2-d coordinate system, and a one-to-one function, there shall be only one global optimal solution. Does it erroneous?
Thanks in Advance.
If you mean one-to-one function from $\mathbb{R}$ to $\mathbb{R} $ then there's no global minimum, because for every $N>0$ there's a point $x$ where $f(x) = -N$.
If you don't mean one-to-one function from $\mathbb{R}$ to $\mathbb{R}$, consider $e^{-x}$: is one-to-one function from $\mathbb{R} $ to $(0, \infty)$ without global minimum.
As you mean one-to-one function, two or more global minumums are not possible, because if $a$ and $b$ are points of global minimum then $f(a)=f(b)$.