Sparse sums of squares on finite abelian groups and improved semidefinite lifts
Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares...
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| Veröffentlicht in: | Mathematical programming Jg. 160; H. 1-2; S. 149 - 191 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.11.2016
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0025-5610, 1436-4646 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | Let
G
be a finite abelian group. This paper is concerned with nonnegative functions on
G
that are
sparse
with respect to the Fourier basis. We establish combinatorial conditions on subsets
S
and
T
of Fourier basis elements under which nonnegative functions with Fourier support
S
are sums of squares of functions with Fourier support
T
. Our combinatorial condition involves constructing a chordal cover of a graph related to
G
and
S
(the Cayley graph
Cay
(
G
^
,
S
)
) with maximal cliques related to
T
. Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of
G
(the characters of
G
). We apply our general result to two examples. First, in the case where
G
=
Z
2
n
, by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in
n
binary variables is a sum of squares of functions of degree at most
n
/
2
, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree
d
on
Z
N
(when
d
divides
N
). By constructing a particular chordal cover of the
d
th power of the
N
-cycle, we prove that any such function is a sum of squares of functions with at most
3
d
log
(
N
/
d
)
nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in
R
2
d
with
N
vertices can be expressed as a projection of a section of the cone of positive semidefinite matrices of size
3
d
log
(
N
/
d
)
. Putting
N
=
d
2
gives a family of polytopes in
R
2
d
with linear programming extension complexity
Ω
(
d
2
)
and semidefinite programming extension complexity
O
(
d
log
(
d
)
)
. To the best of our knowledge, this is the first explicit family of polytopes
(
P
d
)
in increasing dimensions where
xc
PSD
(
P
d
)
=
o
(
xc
LP
(
P
d
)
)
, where
xc
PSD
and
xc
LP
are respectively the SDP and LP extension complexity. |
|---|---|
| Bibliographie: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 0025-5610 1436-4646 |
| DOI: | 10.1007/s10107-015-0977-z |