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I want to find pairs of two functions on two variables $f_1(x,y)$ and $f_2(x,y)$, such that each one can be minimized on one variable ($f_1$ needs to have a minimum on the variable $x$ and $f_2$ needs to have a minimum on $y$) but in such a way that their gradients are bounded, and also their graphs are bounded, and such that the minima of each happen in distinct points $(\bar{x},\bar{y})$.

For instance, if I take $f_1(x,y) = (x-y)^2$, the minimum happens when $x=y$, and if I take $f_2(x,y) = (x-(y-1))^2$, the minimum happens when $x=y-1$. The minima don't have intersection, but the functions and gradient are not bounded. I tried doing something involving sine and cosine but the problem is that there too many minima, I'd rather them have only one, or at least a global minimum...

Are there any examples satisfying all the conditions?

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  • $\begingroup$ What does "limited" mean in this context? Did you mean to say "bounded"? If so, maybe a function like $\exp(-x^2)$ does what you want, as it has a unique critical point and the derivative is obviously bounded. of course, you'll have to modify it slightly to get the sort of functions you specified. $\endgroup$
    – lulu
    Oct 13 at 23:45
  • $\begingroup$ Ah, yes. I meant bounded, thx. While exp(-x^2) is bounded, it does not admit a point on whih it attains the minimum, tho... $\endgroup$
    – karlabos
    Oct 14 at 0:47
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    $\begingroup$ It has a unique critical point, which is all that matters. Trivial modifications are needed to match your criteria (for instance, look at $-\exp(-x^2)$ instead). $\endgroup$
    – lulu
    Oct 14 at 11:12

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