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From: beet on 5 May 2010 20:14
Anyone please help :)

I want to know if it is true that for a piecewise linear function g on
a compact set X, the subgradients of any two connected pieces points
to the similar directions (except for those touch the minimum) in the
sense that (g1.*g2)>0? or something like (g1-g2).*(x1-x2) >= 0?

Sorry I am an engineering student, please help.

Thanks a lot,

Beet
From: Ray Vickson on 5 May 2010 20:30
On May 5, 5:14 pm, beet <whh...(a)gmail.com> wrote:
> Anyone please help :)
>
> I want to know if it is true that for a piecewise linear function g on
> a compact set X, the subgradients of any two connected pieces points
> to the similar directions (except for those touch the minimum) in the
> sense that (g1.*g2)>0? or something like (g1-g2).*(x1-x2) >= 0?
>
> Sorry I am an engineering student, please help.

By "connected", do you mean "adjacent"? I will assume so. The answer
is NO: you can find adjacent planes on the graph of y = f(x1,x2) that
point "oppositely", yet the line joining the planes does not pass
through the minimum---in fact, the minimum can be in a plane different
from either of the two. For a specific example, suppose we slide the
vertex of the graph of y = |x2| up along the line y = x1 for x1 > 0;
that is, at x1 > 0 the graph of y = f(x1,x2) is y = x1 + |x2|. For x1
< 0 slide the vertex of the graph of y = |x2| down along the line y =
x1/2; that is, for x1 < 0 the graph of y is y = x1/2 + |x2|. In the
region -1 <= x1 <= 1, 1- <= x2 <= 1 the function f(x1,x2) is piecewise-
linear convex with minimum at x1 = -1, x2 = 0. However, the first two
planes y = x1+x2 and y = x1-x2 (for x1 >= 0) do not touch the
minimum.

R.G. Vickson

>
> Thanks a lot,
>
> Beet

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