Cuts and semidefinite liftings for the complex cut polytope
We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices $$xx^\textrm{H}$$ x x H , where the elements of $$x \in \mathbb {C}^n$$ x ∈ C n are m th unit roots. These polytopes have applications in MAX-3-CUT, digital communication technology, angular synchronization and more g...
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| Published in: | Mathematical programming |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
09.10.2024
|
| ISSN: | 0025-5610, 1436-4646 |
| Online Access: | Get full text |
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| Summary: | We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices
$$xx^\textrm{H}$$
x
x
H
, where the elements of
$$x \in \mathbb {C}^n$$
x
∈
C
n
are
m
th unit roots. These polytopes have applications in MAX-3-CUT, digital communication technology, angular synchronization and more generally, complex quadratic programming. For
$$m=2$$
m
=
2
, the complex cut polytope corresponds to the well-known cut polytope. We generalize valid cuts for this polytope to cuts for any complex cut polytope with finite
$$m>2$$
m
>
2
and provide a framework to compare them. Further, we consider a second semidefinite lifting of the complex cut polytope for
$$m=\infty $$
m
=
∞
. This lifting is proven to be equivalent to other complex Lasserre-type liftings of the same order proposed in the literature, while being of smaller size. Our theoretical findings are supported by numerical experiments on various optimization problems. |
|---|---|
| ISSN: | 0025-5610 1436-4646 |
| DOI: | 10.1007/s10107-024-02147-3 |