# Sparse sums of squares on finite abelian groups and improved semidefinite lifts

@article{Fawzi2016SparseSO, title={Sparse sums of squares on finite abelian groups and improved semidefinite lifts}, author={Hamza Fawzi and James Saunderson and Pablo A. Parrilo}, journal={Mathematical Programming}, year={2016}, volume={160}, pages={149-191} }

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 $${\mathcal {S}}$$S and $${\mathcal {T}}$$T of Fourier basis elements under which nonnegative functions with Fourier support $${\mathcal {S}}$$S are sums of squares of functions with Fourier support $${\mathcal {T}}$$T. Our combinatorial condition involves constructing a chordal cover of a graph related to G…

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## References

SHOWING 1-10 OF 26 REFERENCES

### Positive semidefinite rank

- MathematicsMath. Program.
- 2015

The main mathematical properties of psd rank are surveyed, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.

### Sums of squares on the hypercube

- Mathematics
- 2014

Let X be a finite set of points in $${\mathbb {R}}^n$$Rn. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of…

### Small Extended Formulations for Cyclic Polytopes

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2015

Through Yannakakis’s factorization theorem, these factorizations yield small-size extended formulations for cyclic polytopes of dimension 3, which are used as base case to construct small-rank nonnegative factorizations of the slack matrices of higher-dimensional cyclicpolytopes.

### Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

- MathematicsSIAM J. Optim.
- 2015

A representation-theoretic framework is presented to study equivariant PSD lifts of a certain class of symmetric polytopes known as orbitopes which respect the symmetries of the polytope.

### Equivariant Semidefinite Lifts of Regular Polygons

- MathematicsMath. Oper. Res.
- 2017

This paper shows that one can construct an equivariant psd lift of the regular 2^n-gon of size 2n-1, which is exponentially smaller than the psd Lift of the sum-of-squares hierarchy, and proves that the construction is essentially optimal.

### Fourier analysis on finite groups and applications

- Mathematics
- 1999

Introduction Cast of characters Part I: 1. Congruences and the quotient ring of the integers mod n 1.2 The discrete Fourier transform on the finite circle 1.3 Graphs of Z/nZ, adjacency operators,…

### Global Optimization with Polynomials and the Problem of Moments

- MathematicsSIAM J. Optim.
- 2001

It is shown that the problem of finding the unconstrained global minimum of a real-valued polynomial p(x): R n to R, in a compact set K defined byPolynomial inequalities reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems.

### On the existence of convex decompositions of partially separable functions

- MathematicsMath. Program.
- 1984

In the sparse case, when eachNi is spanned by Cartesian basis vectors, it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a (permuted) LDLT factorization without fill-in.

### Neighborly and cyclic polytopes

- Mathematics
- 1963

Introduction. Let S be a finite set of points in n-space. A pair of points p and q of S are called neighbors if the segment joining them is an edge of the convex polytope spanned by S. Some years ago…

### Lifts of Convex Sets and Cone Factorizations

- MathematicsMath. Oper. Res.
- 2013

This paper addresses the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone and shows that the existence of a lift of a conveX set to a cone is equivalent to theexistence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual.