Can the Subdifferential of a Non-Convex Function at a Point be Computed?

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Homework Help Overview

The discussion revolves around the computation of the subdifferential of a non-convex function defined as f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0. The original poster is seeking guidance on how to approach this problem, particularly at the point x = 0.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the definition of subdifferentiation and question its applicability to the given non-convex function. The original poster expresses uncertainty about the process and whether a direct formula exists. Others inquire about the convexity of the function and its implications for finding the subdifferential at zero.

Discussion Status

The discussion is ongoing, with participants sharing definitions and exploring the concept of subdifferentiation. Some guidance has been provided regarding the nature of convexity and its relevance to the problem, but no consensus has been reached on how to compute the subdifferential for the specific function in question.

Contextual Notes

Participants note that the function is not convex, which raises questions about the feasibility of finding the subdifferential at the origin. The original poster references an example of a convex function to contrast with their own, indicating a potential misunderstanding of the concepts involved.

braindead101
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Define f:R->R by
f(x) = { x^2 sin(1/x) x!=0, 0 x=0
Compute subdifferential f(0)

I went through my notes on subdifferentiation and still do not have a clue how to do this process, is there a formula to directly do this? any help would be greatly appreciated.
 
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On the entire of the world wide web, the first result on "subdifferentiation" comes from this thread. What exactly is subdifferentiation?
 
Ok with that definition:

What makes you (the OP) think that we can find the subderiavative at zero? Is the function convex...
 
the function is not convex. but it is an assignment question so it must be doable some how.
so i looked at the wiki you sent me and saw the example. but then i guess it is not the same
as my question as that function is convex and mine is not.
Example i am referring to:
Consider the function f(x)=|x| which is convex. Then, the subdifferential at the origin is the interval [−1, 1].

But with that said, I graphed the x^2sin(1/x) function to try to see the behaviour around 0, and both sides are approaching 0 and the slope seems to be also 0.. but i don't know if this is how to actually do it.

what are your thoughts after that wiki read?
 

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