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  • 0 posts edited
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  • 7 votes cast
Mar
31
awarded  Tumbleweed
Mar
24
asked Does every boundary segment of a convex polyhedron lies on one of its faces?
Mar
24
comment Are these equivalent definitions of faces of convex sets?
@timhortons Thanks a lot.
Mar
23
revised Are these equivalent definitions of faces of convex sets?
added 3 characters in body
Mar
23
revised Are these equivalent definitions of faces of convex sets?
added 2 characters in body
Mar
23
asked Are these equivalent definitions of faces of convex sets?
Mar
23
asked Are these inconsistent definitions of extreme subset of convex set?
Oct
3
comment Will continuous extension preserve strict convexity?
This function is not even convex. You can consider $g(x)=h(x,x)=(x^2-1)^2$.
Oct
3
asked Will continuous extension preserve strict convexity?
Sep
21
accepted Is the limit point of this sequence unique?
Sep
21
comment Is the limit point of this sequence unique?
Great. I'd better pick up a book and review some basics.
Sep
21
comment Is the limit point of this sequence unique?
Thanks for your quick response. Please see my edit.
Sep
21
revised Is the limit point of this sequence unique?
added 48 characters in body
Sep
21
asked Is the limit point of this sequence unique?
Sep
15
comment normal cone to sublevel set
Without the assumption that ${\rm cone}\,\partial f(\bar{x})$ is closed, why is the inclusion $N_C(\bar{x})\subset{\rm cone}\,\partial f(\bar{x})$ equivalent to $(N_C(\bar{x}))^{\circ}\supset({\rm cone}\,\partial f(\bar{x}))^{\circ}$?
Sep
15
comment normal cone to sublevel set
This is indeed an exercise in Borwein's book: Convex analysis and nonlinear optimization. Although it is an exercise, I think it is very useful to relate many stuff together. As I mentioned, it appears in many other textbooks as a theorem in a bit different form.
Sep
15
comment normal cone to sublevel set
Moreover, I do not think a direction polar to all subgradients has to be a descent direction. Consider $f(x,y)=\frac{1}{2}(x^2+y^2)$ at $(1,0)$. The gradient is $(1,0)$. The direction $(0,1)$ is polar to $(1,0)$ and belong to the tangent cone. But it is indeed an ascent direction. Please let me know if I miss anything. Thanks.
Sep
15
comment normal cone to sublevel set
Hi, gerw: could you please provide more detail about the last inequality? By the way, I think you also make use of the fact that ${\rm cone}\,\partial f(\bar{x})$ is closed. Thus, it is the same as $({\rm cone}\,\partial f(\bar{x}))^{\circ\circ}$. This is due to $0\notin\partial f(\bar{x})$. Am I right? Thanks a lot.
Sep
13
asked normal cone to sublevel set
Sep
1
accepted Is the smallest singular value able to measure the similarity between two matrices?