#18392: Add is_solid() and is_proper() for Polyhedral cones
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Reporter: mjo | Owner:
Type: enhancement | Status: positive_review
Priority: major | Milestone: sage-6.7
Component: geometry | Resolution:
Keywords: | Merged in:
Authors: Michael Orlitzky | Reviewers: Andrey Novoseltsev
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/mjo/ticket/18392 | c7e16d9e9ca8a29c24d40a83693109618a0e3415
Dependencies: | Stopgaps:
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Comment (by mjo):
So, I'm interested in computing the Lyapunov (or bilinearity) rank of
cones. The best reference I have is
http://www.math.umbc.edu/~gowda/papers/trGOW13-01.pdf, but basically, I
want to know the dimension of the space of all transformations `L` with
`<Lx,s> = 0` for all `(x,s)` in the complementarity set of the cone `K`.
If the dimension is large, we can do a trick like we do in the linear
complementarity problem where we rewrite the condition `<x,s> = 0` as
`x_{i}s_{i} = 0` (for i=1,...n).
In general this seems very hard, but for polyhedral cones, it should be
easy (if slow) to attack combinatorially: there are only so many elements
in the complementarity set of `K` that we need to look at since the cone
is finitely generated. The first problem I ran into is that in the
literature so far, the cone `K` is always assumed to be proper. I planned
to throw an error if `K` was not proper, but dropping that assumption has
given me some interesting questions to play with over the last few days.
In that sense I'm using them to research the Lyapunov rank of non-proper
cones, but I'm not one of the people who actually tries to solve
optimization problems =)
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Ticket URL: <http://trac.sagemath.org/ticket/18392#comment:6>
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