 Apr 2018

wiki.c2.com wiki.c2.com

ConvexHull
In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X. Wikipedia
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 Jul 2016

arxiv.org arxiv.org

Ax=b
Apparently \(Ax = b\) is not required, but is used as a technical prop in subsequent proof construction.
If no linear constraints are found, either A, b can be viewed as zero, or can be viewed as the smallest affine set that includes S. In both cases, this effectively makes the constraint trivial.

A
How does one go about choosing (a good) A, especially since it seems to not be necessary to have a nontrivial (A,b) pair?

x()=2Feas, that is,1(x)< , establishing the claim
So it appears that we don't need to get feasible points at each iteration?
Answer: correct, this is pointed out later in a couple of pages, where it is stated \(x\) need not be feasible, but \(x'\) will be.

int(dom(f))
Why isn't \(x \in dom(f)\) sufficient?
Ah, I think it is sufficient for \(t' = f(x')\), but we also need \(x \in int(dom(f))\) for the subsequently mentioned subdifferentials.

(x0;t0)2bdy(dom(f))
Confirmed that this should be \(bdy(epi(f))\), not \(bdy(dom(f))\)

because(x;t) =1(x)
Why?

a value we now know to be positive and nite
Where is the proof that if \( \alpha_1 (x) \) is infinite, then \( \alpha_2 (x,t) \) is finite?
I now see that it starts at the beginning of page 3, but without any forward reference it seems that it is being skipped.

Approximating2(x;t) is the same as approximating the solutionto() = 0,
Not completely clear to me why this is the case.

Assume eis a known feasible point contained in int(S\dom(f)).
How to pick? Perhaps a reference to strategies, or note that strategies will be discussed later.
