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There is a statement in a thesis I am reading,

The solution of a convex optimization problem is unique, and the global and the local minima are essentially the same.

Is there a proof for it? Why does it hold?

Gigili
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  • This claim is simply false. Also what do you mean by "essentially the same" ? – dohmatob Sep 14 '16 at 07:51
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    I think what may have been meant is that any local minimum is also a global minimum and that a strictly convex objective function has at most one global minimum. See also: http://math.stackexchange.com/questions/337090/if-f-is-strictly-convex-in-a-convex-set-show-it-has-no-more-than-1-minimum – tibL Sep 14 '16 at 08:25

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The statement is not true! For example, the convex problem: $$\min_{x} f(x) ,$$ such that $f(x)$ is given by

$$f(x) = \begin{cases} 0, & \mbox{if } -1 < x < 1 \\ \lvert x\rvert -1 , & \mbox{ otherwise} \end{cases}$$

has infinitely many local/global minima.

Alex Silva
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