# Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

@article{Fawzi2013EquivariantSL, title={Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies}, author={Hamza Fawzi and James Saunderson and Pablo A. Parrilo}, journal={SIAM J. Optim.}, year={2013}, volume={25}, pages={2212-2243} }

A central question in optimization is to maximize (or minimize) a linear function over a given polytope $P$. To solve such a problem in practice one needs a concise description of the polytope $P$. In this paper we are interested in representations of $P$ using the positive semidefinite cone: a positive semidefinite lift (PSD lift) of a polytope $P$ is a representation of $P$ as the projection of an affine slice of the positive semidefinite cone $\mathbf{S}^d_+$. Such a representation allows…

## 25 Citations

### 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.

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It is shown that the cone of $k \times k$ symmetric positive semidefinite matrices has no extended formulation with finitely many LMIs of size less than $k$ and the standard extended formulation of $\Sigma_{n,2d}$ is optimal in terms of the size of the LMIs.

### Lifting for Simplicity: Concise Descriptions of Convex Sets

- MathematicsSIAM Review
- 2022

The connection between the existence of lifts of a convex set and certain structured factorizations of its associated slack operator is explained, and a uniform approach to the construction of spectrahedral lifts of convex sets is described.

### Lower Bounds on the Size of Semidefinite Programming Relaxations

- Mathematics, Computer ScienceSTOC
- 2015

It is proved that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations, and this result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes.

### SPECTRAHEDRAL LIFTS OF CONVEX SETS

- MathematicsProceedings of the International Congress of Mathematicians (ICM 2018)
- 2019

Efficient representations of convex sets are of crucial importance for many algorithms that work with them. It is well-known that sometimes, a complicated convex set can be expressed as the…

### SPECTRAHEDRAL LIFTS OF CONVEX SETS

- Mathematics
- 2018

Efficient representations of convex sets are of crucial importance for many algorithms that work with them. It is well-known that sometimes, a complicated convex set can be expressed as the…

### Semidefinite Descriptions of the Convex Hull of Rotation Matrices

- MathematicsSIAM J. Optim.
- 2015

It is shown that the convex hull of SO(n) is doubly spectrahedral, i.e., both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace.

### Sparse sum-of-squares certificates on finite abelian groups

- Mathematics, Computer Science2015 54th IEEE Conference on Decision and Control (CDC)
- 2015

This paper considers the problem of finding sparse sum-of-squares certificates for functions defined on a finite abelian group G and builds the first explicit family of polytopes in increasing dimensions that have a semidefinite programming description that is vanishingly smaller than any linear programming description.

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

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- 2016

It is proved 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.

### Semidefinite representations with applications in estimation and inference

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## References

SHOWING 1-10 OF 40 REFERENCES

### 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.

### 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.

### Polytopes of Minimum Positive Semidefinite Rank

- MathematicsDiscret. Comput. Geom.
- 2013

This paper shows that the psd rank of a polytope is at least the dimension of the polytopes plus one, and characterize those polytopes whose pSD rank equals this lower bound.

### 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.

### Lower Bounds on the Size of Semidefinite Programming Relaxations

- Mathematics, Computer ScienceSTOC
- 2015

It is proved that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations, and this result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes.

### Semidefinite Descriptions of the Convex Hull of Rotation Matrices

- MathematicsSIAM J. Optim.
- 2015

It is shown that the convex hull of SO(n) is doubly spectrahedral, i.e., both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace.

### A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs

- MathematicsMath. Program.
- 2012

The theta bodies of the vanishing ideal of cycles in a binary matroid are constructed and applied to cuts in graphs, this yields a new hierarchy of semidefinite programming relaxations of the cut polytope of the graph.

### Semidefinite Relaxations for Max-Cut

- MathematicsThe Sharpest Cut
- 2004

It is shown that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the max-cut problem, and the class L t of the graphs G for which Q t (G) is the cut polytope of G is shown to be closed under taking minors.

### Semidefinite Optimization and Convex Algebraic Geometry

- Mathematics
- 2012

This book provides a self-contained, accessible introduction to the mathematical advances and challenges resulting from the use of semidefinite programming in polynomial optimization. This quickly…