$$ \min\limits_{x\in \mathcal{X}} \|x\|_2^2 $$ where $\mathcal{X}\subseteq R^N$ is a closed set but not necessary convex and can be unbounded. Then existence of optimal solution to this problem holds or not?
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Yes, of course, because $\lVert x\rVert_2^2$ is continuous and it diverges to infinity away from $0$. Not necessarily unique, of course. – Apr 24 '19 at 07:17
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What is the space in which you are working? Is it finite dimensional? – Kavi Rama Murthy Apr 24 '19 at 07:44
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Thanks for Saucy and Kavi's help. The problem has solved. The objective is coercive on the closed set $\mathcal{X}$. So it attains its minimum on $\mathcal{X}$. – zly May 04 '19 at 13:10