Can someone give an example of a convex function $f$ on a path-connected compact nonconvex set where some point $c$ is a local minimum with $\nabla f(c)=0$ but not a global minimum. Thus showing that the set being not convex makes finding global minimums harder.
Convex function on nonconvex set and global minimum
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convex-optimization
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0Well presumably the focus would be on set with interior having some meat to it. – 2012-03-11
2 Answers
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Consider the set $A:=[-1,3]\times[-3,3]\ \setminus\ \{(x,y)\ |\ 0<|y|
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0@Rahul Narain: There was a typo in the definition of the isosceles triangle. Thank you. – 2012-03-11
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For example the function $f(x,y) = x^2+y^2$ restricted to the square with vertices $(-1,-2)$, $(5,-2)$, $(5,4)$ and $(-1,4)$ has four local minima but only one global minimum.
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0@user782220 As Rahul Narain commented: The boundary of the square. (Otherwise $f$ would clearly only have a single local minimum.) – 2012-03-11