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 |
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01.11.2016
Springer Nature B.V |
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| Abstract | 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. |
|---|---|
| AbstractList | (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) 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 ... and ... of Fourier basis elements under which nonnegative functions with Fourier support ... are sums of squares of functions with Fourier support ... Our combinatorial condition involves constructing a chordal cover of a graph related to G and ... (the Cayley graph ...) with maximal cliques related to ... 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 ..., 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 ..., establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on ... (when d divides N). By constructing a particular chordal cover of the dth power of the N-cycle, we prove that any such function is a sum of squares of functions with at most ... nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in ... with N vertices can be expressed as a projection of a section of the cone of positive semidefinite matrices of size ... Putting ... gives a family of polytopes in ... with linear programming extension complexity ... and semidefinite programming extension complexity ... To the best of our knowledge, this is the first explicit family of polytopes ... in increasing dimensions where ..., where ... and ... are respectively the SDP and LP extension complexity. 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. |
| Author | Fawzi, Hamza Parrilo, Pablo A. Saunderson, James |
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| Cites_doi | 10.1287/moor.28.4.871.20508 10.1016/0024-3795(88)90240-6 10.1007/s10107-015-0922-1 10.1137/S1052623400366802 10.1016/0022-0000(91)90024-Y 10.1007/BF02612711 10.1016/j.disc.2012.09.015 10.1017/CBO9780511626265 10.1016/0024-3795(84)90207-6 10.1287/moor.1120.0575 10.1137/140966265 10.1002/9781118165621 10.1137/1.9781611972290 10.1007/978-3-642-20807-2_23 10.1007/978-1-4757-3216-0_17 10.1007/s00454-015-9682-1 10.1007/978-1-4613-8431-1 10.1090/pspum/007/0152944 |
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| DOI | 10.1007/s10107-015-0977-z |
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| Snippet | 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... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Let G be a finite abelian group. This paper is concerned with nonnegative functions... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).Let G be a finite abelian group. This paper is concerned with nonnegative functions... |
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| SubjectTerms | Boolean Calculus of Variations and Optimal Control; Optimization Combinatorics Complexity Computer engineering Computer science Construction Decision support systems Decomposition Electrical engineering Fourier analysis Full Length Paper Functions (mathematics) Graphs Laboratories Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Polytopes Semidefinite programming Studies Theoretical |
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| Title | Sparse sums of squares on finite abelian groups and improved semidefinite lifts |
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