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Consider the problem: minimize $f(x)=x_1$ subject to $(x_1-1)^2+x_2^2=1$, $(x_1+1)^2+x_2^2=1.$ Are there any local minimizers? Are there any global minimizers?

Is $(-2,0)$ both the local and global minimizer?

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$f$ is a function of two variables, $x_1$ and $x_2$, even though it depends on $x_1$ only. So minimizers, whatever they are, will be pairs of numbers, like $(x_1,x_2)=(43,-84)$. – user53153 Jan 17 '13 at 21:49

You constrain your function to the intersection of the circles of radius 1 centred at $(-1,0)$ and $(1,0)$, i.e. you minimize the function f over the set $\{(0,0)\}$. Consequently, $(0,0)$ is the only local and global optimiser.

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but the object function is $f(x)=x_1$. $x_1=-2$ is the minimal value of $f(x)=x_1$. – i_a_n Jan 17 '13 at 22:44

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