Showing papers by "Hamza Fawzi published in 2015"

Positive semidefinite rank

Abstract: Let $$M \in \mathbb {R}^{p \times q}$$M?Rp×q be a nonnegative matrix. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices $$A_i, B_j$$Ai,Bj of size $$k \times k$$k×k such that $$M_{ij} = {{\mathrm{trace}}}(A_i B_j)$$Mij=trace(AiBj). The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedra and information-theoretic applications. In this paper we develop and survey the main mathematical properties of psd rank, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.

Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

Abstract: 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 linear optimization problems over $P$ to be written as semidefinite programs of size $d$. Such representations can be beneficial in practice when $d$ is much smaller than the number of facets of the polytope $P$. In this paper we are concerned with so-called equivariant PSD lifts (also known as symmetric PSD lifts) which respect the symmetries of the polytope $P$. We present a representation-theoretic framework to study equivariant PSD lifts of a certain class of symmetric polytopes known as orbitopes. Our main result...

Sparse sum-of-squares certificates on finite abelian groups

Abstract: Sums-of-squares techniques have played an important role in optimization and control. One question that has attracted a lot of attention is to exploit sparsity in order to reduce the size of sum-of-squares programs. In this paper we consider the problem of finding sparse sum-of-squares certificates for functions defined on a finite abelian group G. In this setting the natural basis over which to measure sparsity is the Fourier basis of G (also called the basis of characters of G). We establish combinatorial conditions on subsets S and τ of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support τ. Our combinatorial condition involves constructing a chordal cover of a graph related to G and S with maximal cliques related to τ. These techniques allow us to show that any nonnegative quadratic function in binary variables is a sum of squares of functions of degree at most ⌈n/2⌉, resolving a conjecture of Laurent [11]. They also allow us to show that any nonnegative function of degree d on G = ℤN has a sum-of-squares certificate supported on at most 3d log(N/d) Fourier basis elements. By duality this construction yields the first explicit family of polytopes in increasing dimensions that have a semidefinite programming description that is vanishingly smaller than any linear programming description.

Lower bounds on nonnegative rank via nonnegative nuclear norms

Abstract: The nonnegative rank of an entrywise nonnegative matrix $$A \in \mathbb {R}^{m \times n}_+$$A?R+m×n is the smallest integer $$r$$r such that $$A$$A can be written as $$A=UV$$A=UV where $$U \in \mathbb {R}^{m \times r}_+$$U?R+m×r and $$V \in \mathbb {R}^{r \times n}_+$$V?R+r×n are both nonnegative. The nonnegative rank arises in different areas such as combinatorial optimization and communication complexity. Computing this quantity is NP-hard in general and it is thus important to find efficient bounding techniques especially in the context of the aforementioned applications. In this paper we propose a new lower bound on the nonnegative rank which, unlike most existing lower bounds, does not solely rely on the matrix sparsity pattern and applies to nonnegative matrices with arbitrary support. The idea involves computing a certain nuclear norm with nonnegativity constraints which allows to lower bound the nonnegative rank, in the same way the standard nuclear norm gives lower bounds on the standard rank. Our lower bound is expressed as the solution of a copositive programming problem and can be relaxed to obtain polynomial-time computable lower bounds using semidefinite programming. We compare our lower bound with existing ones, and we show examples of matrices where our lower bound performs better than currently known ones.

Sparse sum-of-squares certificates on finite abelian groups

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($\hat{G}$,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. We apply our general result to two examples. First, in the case where $G = \mathbb{Z}_2^n$, 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 $\lceil n/2 \rceil$, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on $\mathbb{Z}_N$ (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 $3d\log(N/d)$ nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in $\mathbb{R}^{2d}$ with N vertices can be expressed as a projection of a section of the cone of psd matrices of size $3d\log(N/d)$. Putting $N=d^2$ gives a family of polytopes $P_d \subset \mathbb{R}^{2d}$ with LP extension complexity $\text{xc}_{LP}(P_d) = \Omega(d^2)$ and SDP extension complexity $\text{xc}_{PSD}(P_d) = O(d\log(d))$. To the best of our knowledge, this is the first explicit family of polytopes in increasing dimensions where $\text{xc}_{PSD}(P_d) = o(\text{xc}_{LP}(P_d))$.

Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

Abstract: 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 linear optimization problems over $P$ to be written as semidefinite programs of size $d$. Such representations can be beneficial in practice when $d$ is much smaller than the number of facets of the polytope $P$. In this paper we are concerned with so-called equivariant PSD lifts (also known as symmetric PSD lifts) which respect the symmetries of the polytope $P$. We present a representation-theoretic framework to study equivariant PSD lifts of a certain class of symmetric polytopes known as orbitopes. Our main result...

Lieb's concavity theorem, matrix geometric means, and semidefinite optimization

Abstract: A famous result of Lieb establishes that the map $(A,B) \mapsto \text{tr}\left[K^* A^{1-t} K B^t\right]$ is jointly concave in the pair $(A,B)$ of positive definite matrices, where $K$ is a fixed matrix and $t \in [0,1]$. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational $t \in [0,1]$. Our construction makes use of a semidefinite formulation of weighted matrix geometric means. We provide an implementation of our constructions in Matlab.

Sparse sum-of-squares certicates on

Abstract: Let G be a nite 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 subsetsS andT of Fourier basis elements under which nonnegative functions with Fourier supportS are sums of squares of functions with Fourier supportT . Our combinatorial condition involves constructing a chordal cover of a graph related to G andS (the Cayley graph Cay( b G;S)) with maximal cliques related toT . Our result relies on two main ingredients: the decomposition of sparse positive semidenite 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 n , by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form inn binary variables is a sum of squares of functions of degree at mostdn=2e, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on ZN (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 3d log(N=d) nonzero Fourier coecients. Dually this shows that a certain cyclic polytope in R 2d with N vertices can be expressed as a projection of a section of the