Timeline for For a convex function, can subgradients be formed from finite convex combinations of gradients?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 13, 2017 at 16:36 | comment | added | Anton Petrunin | @Mehrdad, no, there are no interesting examples; this is easy to check. | |
| Mar 13, 2017 at 5:56 | comment | added | user541686 | @AntonPetrunin: If I understand this example correctly, the subgradient (1,0) can be approximated arbitrarily well by a convex combination of gradients in a ball around (0,0)... so we might as well call it a subgradient, right? If so, it seems a bit like telling someone that $\max x : x < 1$ doesn't exist... which is true, but not all that interesting. Are there any interesting counterexamples, or are they all correct if we're willing to take reasonable limits as needed? | |
| Aug 6, 2016 at 23:38 | comment | added | Austin Bren | Thanks for the responses. Does this fact go for when we have gradients almost everywhere on the ball (and subgradients on the non-differentiable points)? Also, any citations or literature to go with this? | |
| Aug 6, 2016 at 22:32 | comment | added | Anton Petrunin | @TravisHartford, in this case it is true. In fact if the gradient is defined at all points of a small sphere then any subgradient inside is a convex combination of at most $n+1$ of gradients on the sphere. | |
| Aug 6, 2016 at 22:11 | comment | added | Austin Bren | I wonder if there is also an example where all $x$ within the unit ball are defined for the function. I feel that this example only works (it is a valid counterexample) because there are undefined points in the ball. | |
| Aug 6, 2016 at 22:09 | vote | accept | Austin Bren | ||
| Aug 6, 2016 at 22:05 | vote | accept | Austin Bren | ||
| Aug 6, 2016 at 22:09 | |||||
| S Aug 6, 2016 at 20:45 | history | suggested | Fabian Wirth | CC BY-SA 3.0 |
Fixed grammar and took care of a typo.
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| Aug 6, 2016 at 20:24 | review | Suggested edits | |||
| S Aug 6, 2016 at 20:45 | |||||
| Aug 6, 2016 at 14:34 | history | answered | Anton Petrunin | CC BY-SA 3.0 |