arXiv subjects

Monique Laurent

Publications and source records attributed to Monique Laurent.

At least 19 recordsLinked to original sources

arXiv2026

The Algebraic Boundary of Graph Elliptopes

This paper studies the algebraic boundary of the elliptope $\mathcal{E}(G)$ of a graph $G$. In particular, we completely characterize the algebraic boundary of $\mathcal{E}(G)$ when $G$ is cycle completable. In this case, the boundary is a union of determinantal hypersurfaces and Lissajous varieties, i.e., images of rational linear subspaces under the coordinatewise cosine map. As an application, we show that the algebraic boundary of $\mathcal{E}(G)$ is disjoint from its interior precisely when $\mathcal{E}(G)$ is a spectrahedron or, equivalently, when $G$ is a chordal graph. A central ingredient for the defining equation of the boundary hypersurface is the cycle polynomial, which captures the algebraic boundary of the elliptope $\mathcal{E}(C_n)$ of the $n$-th cycle graph $C_n$. We show that the cycle polynomial of $C_n$ is the resultant of two smaller cycle polynomials. Via this result, Sylvester's determinantal formula offers an inductive method for computing the cycle polynomial which mirrors a geometric property of metric polytopes. We also determine the degree of the homogeneous cycle polynomial, settling an open question of Sturmfels and Uhler (2010).

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arXiv2025

Semidefinite hierarchies for diagonal unitary invariant bipartite quantum states

We investigate questions about the cone $\mathrm{SEP}_n$ of separable bipartite states, consisting of the Hermitian matrices acting on $\mathbb{C}^n\otimes\mathbb{C}^n$ that can be written as conic combinations of rank one matrices of the form $xx^*\otimes yy^*$ with $x,y\in\mathbb{C}^n$. Bipartite states that are not separable are said to be entangled. Detecting quantum entanglement is a fundamental task in quantum information and a hard computational problem. We explore the Doherty-Parrilo-Spedalieri (DPS) hierarchy of semidefinite conic approximations for $\mathrm{SEP}_n$ when the bipartite states have some additional structural properties: first, (i) for states with diagonal unitary invariance, and second (ii) for states with Bose symmetry. In case (i) we show that the DPS hierarchy can be block diagonalized, which, combining with its moment reformulation, leads to a substantially more efficient implementation. In case (ii), we give a characterization of the dual hierarchy, in terms of sums of squares of Hermitian complex polynomials, extending a known result in the generic case. It turns out that the completely positive cone $\mathrm{CP}_n$, its dual cone $\mathrm{COP}_n$, and their sums-of-squares based conic approximations $\mathcal{K}^{(t)}_n$, play a central role in these two settings (i),(ii). We clarify these connections and test the block diagonal relaxations on classes of examples.

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arXiv2025

An Overview of Convergence Rates for Sum of Squares Hierarchies in Polynomial Optimization

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex) optimization problems. Various hierarchies of (lower and upper) bounds have been introduced, having the remarkable property that they converge asymptotically to the global minimum. These bounds exploit algebraic representations of positive polynomials in terms of sums of squares and can be computed using semidefinite optimization. Our focus lies in the performance analysis of these hierarchies of bounds, namely, in how far the bounds are from the global minimum as the degrees of the sums of squares they involve tend to infinity. We present the main state-of-the-art results and offer a gentle introductory overview over the various techniques that have been recently developed to establish them, stemming from the theory of orthogonal polynomials, approximation theory, Fourier analysis, and more.

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arXiv2024

Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems

We revisit the convergence analysis of two approximation hierarchies for polynomial optimization on the unit sphere. The first one is based on the moment-sos approach and gives semidefinite bounds for which Fang and Fawzi (2021) showed an analysis in $O(1/r^2)$ for the r-th level bound, using the polynomial kernel method. The second hierarchy was recently proposed by Lovitz and Johnston (2023) and gives spectral bounds for which they show a convergence rate in $O(1/r)$, using a quantum de Finetti theorem of Christandl et al. (2007) that applies to complex Hermitian matrices with a "double" symmetry. We investigate links between these approaches, in particular, via duality of moments and sums of squares. Our main results include showing that the spectral bounds cannot have a convergence rate better than $O(1/r^2)$ and that they do not enjoy generic finite convergence. In addition, we propose alternative performance analyses that involve explicit constants depending on intrinsic parameters of the optimization problem. For this we develop a novel "banded" real de Finetti theorem that applies to real matrices with "double" symmetry. We also show how to use the polynomial kernel method to obtain a de Finetti type result in $O(1/r^2)$ for real maximally symmetric matrices, improving an earlier result in $O(1/r)$ of Doherty and Wehner (2012).

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arXiv2024

Finite convergence of sum-of-squares hierarchies for the stability number of a graph

We investigate a hierarchy of semidefinite bounds $\vartheta^{(r)}(G)$ for the stability number $α(G)$ of a graph $G$, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [{\em SIAM J. Optim.} 12 (2002), pp.875--892], who conjectured convergence to $α(G)$ in $r=α(G)-1$ steps. Even the weaker conjecture claiming finite convergence is still open. We establish links between this hierarchy and sum-of-squares hierarchies based on the Motzkin-Straus formulation of $α(G)$, which we use to show finite convergence when $G$ is acritical, i.e., when $α(G\setminus e)=α(G)$ for all edges $e$ of $G$. This relies, in particular, on understanding the structure of the minimizers of the Motzkin-Straus formulation and showing that their number is finite precisely when $G$ is acritical. Moreover we show that these results hold in the general setting of the weighted stable set problem for graphs equipped with positive node weights. In addition, as a byproduct we show that deciding whether a standard quadratic program has finitely many minimizers does not admit a polynomial-time algorithm unless P=NP.

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arXiv2024

Semidefinite approximations for bicliques and biindependent pairs

We investigate some graph parameters dealing with biindependent pairs $(A,B)$ in a bipartite graph $G=(V_1\cup V_2,E)$, i.e., pairs $(A,B)$ where $A\subseteq V_1$, $B\subseteq V_2$ and $A\cup B$ is independent. These parameters also allow to study bicliques in general graphs. When maximizing the cardinality $|A\cup B|$ one finds the stability number $α(G)$, well-known to be polynomial-time computable. When maximizing the product $|A|\cdot |B|$ one finds the parameter $g(G)$, shown to be NP-hard by Peeters (2003), and when maximizing the ratio $|A|\cdot |B|/|A\cup B|$ one finds $h(G)$, introduced by Vallentin (2020) for bounding product-free sets in finite groups. We show that $h(G)$ is an NP-hard parameter and, as a crucial ingredient, that it is NP-complete to decide whether a bipartite graph $G$ has a balanced maximum independent set. These hardness results motivate introducing semidefinite programming bounds for $g(G)$, $h(G)$, and $α_{\text{bal}}(G)$ (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lovász $\vartheta$-number, a well-known semidefinite bound on $α(G)$. In addition we formulate closed-form eigenvalue bounds and we show relationships among them as well as with earlier spectral parameters by Hoffman, Haemers (2001) and Vallentin (2020).

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arXiv2023

Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks

We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. This sparsity exploits the presence of equality constraints requiring the measure to be supported on the variety of an ideal generated by bilinear monomials modeled by an associated graph. We show that this enables an equivalent sparse reformulation of the GMP, where the single (high dimensional) measure variable is replaced by several (lower-dimensional) measure variables supported on the maximal cliques of the graph. We explore the resulting hierarchies of moment-based relaxations for the original dense formulation of GMP and this new, equivalent ideal-sparse reformulation, when applied to the problem of bounding nonnegative- and completely positive matrix factorization ranks. We show that the ideal-sparse hierarchies provide bounds that are at least as good (and often tighter) as those obtained from the dense hierarchy. This is in sharp contrast to the situation when exploiting correlative sparsity, as is most common in the literature, where the resulting bounds are weaker than the dense bounds. Moreover, while correlative sparsity requires the underlying graph to be chordal, no such assumption is needed for ideal-sparsity. Numerical results show that the ideal-sparse bounds are often tighter and much faster to compute than their dense analogs.

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arXiv2023

Copositive matrices, sums of squares and the stability number of a graph

This chapter investigates the cone of copositive matrices, with a focus on the design and analysis of conic inner approximations for it. These approximations are based on various sufficient conditions for matrix copositivity, relying on positivity certificates in terms of sums of squares of polynomials. Their application to the discrete optimization problem asking for a maximum stable set in a graph is also discussed. A central theme in this chapter is understanding when the conic approximations suffice for describing the full copositive cone, and when the corresponding bounds for the stable set problem admit finite convergence.

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arXiv2023

An effective version of Schmüdgen's Positivstellensatz for the hypercube

Let $S \subseteq \mathbb{R}^n$ be a compact semialgebraic set and let $f$ be a polynomial nonnegative on $S$. Schmüdgen's Positivstellensatz then states that for any $η> 0$, the nonnegativity of $f + η$ on $S$ can be certified by expressing $f + η$ as a conic combination of products of the polynomials that occur in the inequalities defining $S$, where the coefficients are (globally nonnegative) sum-of-squares polynomials. It does not, however, provide explicit bounds on the degree of the polynomials required for such an expression. We show that in the special case where $S = [-1, 1]^n$ is the hypercube, a Schmüdgen-type certificate of nonnegativity exists involving only polynomials of degree $O(1 / \sqrtη)$. This improves quadratically upon the previously best known estimate in $O(1/η)$. Our proof relies on an application of the polynomial kernel method, making use in particular of the Jackson kernel on the interval $[-1, 1]$.

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arXiv2022

On the Exactness of Sum-of-Squares Approximations for the Cone of $5\times 5$ Copositive Matrices

We investigate the hierarchy of conic inner approximations $\mathcal{K}^{(r)}_n$ ($r\in \mathbb{N}$) for the copositive cone $\text{COP}_n$, introduced by Parrilo (Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD Thesis, California Institute of Technology, 2001). It is known that $\text{COP}_4=\mathcal{K}^{(0)}_4$ and that, while the union of the cones $\mathcal{K}^{(r)}_n$ covers the interior of $\text{COP}_n$, it does not cover the full cone $\text{COP}_n$ if $n\geq 6$. Here we investigate the remaining case $n=5$, where all extreme rays have been fully characterized by Hildebrand (The extreme rays of the 5 $\times$ 5 copositive cone. Linear Algebra and its Applications, 437(7):1538--1547, 2012). We show that the Horn matrix $H$ and its positive diagonal scalings play an exceptional role among the extreme rays of $\text{COP}_5$. We show that equality $\text{COP}_5=\bigcup_{r\geq 0} \mathcal{K}^{(r)}_5$ holds if and only if any positive diagonal scaling of $H$ belongs to $\mathcal{K}^{(r)}_5$ for some $r\in \mathbb{N}$. As a main ingredient for the proof, we introduce new Lasserre-type conic inner approximations for $\text{COP}_n$, based on sums of squares of polynomials. We show their links to the cones $\mathcal{K}^{(r)}_n$, and we use an optimization approach that permits to exploit finite convergence results on Lasserre hierarchy to show membership in the new cones.

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arXiv2022

Sum-of-squares hierarchies for binary polynomial optimization

We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial $f$ over the boolean hypercube $\mathbb{B}^{n}=\{0,1\}^n$. This hierarchy provides for each integer $r \in \mathbb{N}$ a lower bound $f_{(r)}$ on the minimum $f_{\min}$ of $f$, given by the largest scalar $λ$ for which the polynomial $f - λ$ is a sum-of-squares on $\mathbb{B}^{n}$ with degree at most $2r$. We analyze the quality of these bounds by estimating the worst-case error $f_{\min} - f_{(r)}$ in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed $t \in [0, 1/2]$, we can show that this worst-case error in the regime $r \approx t \cdot n$ is of the order $1/2 - \sqrt{t(1-t)}$ as $n$ tends to $\infty$. Our proof combines classical Fourier analysis on $\mathbb{B}^{n}$ with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds $f_{(r)}$ and another hierarchy of upper bounds $f^{(r)}$, for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the $q$-ary cube $(\mathbb{Z}/q\mathbb{Z})^{n}$.

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arXiv2021

Bounding the separable rank via polynomial optimization

We investigate questions related to the set $\mathcal{SEP}_d$ consisting of the linear maps $ρ$ acting on $\mathbb{C}^d\otimes \mathbb{C}^d$ that can be written as a convex combination of rank one matrices of the form $xx^*\otimes yy^*$. Such maps are known in quantum information theory as the separable bipartite states, while nonseparable states are called entangled. In particular we introduce bounds for the separable rank $\mathrm{rank_{sep}}(ρ)$, defined as the smallest number of rank one states $xx^*\otimes yy^*$ entering the decomposition of a separable state $ρ$. Our approach relies on the moment method and yields a hierarchy of semidefinite-based lower bounds, that converges to a parameter $τ_{\mathrm{sep}}(ρ)$, a natural convexification of the combinatorial parameter $\mathrm{rank_{sep}}(ρ)$. A distinguishing feature is exploiting the positivity constraint $ρ-xx^*\otimes yy^* \succeq 0$ to impose positivity of a polynomial matrix localizing map, the dual notion of the notion of sum-of-squares polynomial matrices. Our approach extends naturally to the multipartite setting and to the real separable rank, and it permits strengthening some known bounds for the completely positive rank. In addition, we indicate how the moment approach also applies to define hierarchies of semidefinite relaxations for the set $\mathcal{SEP}_d$ and permits to give new proofs, using only tools from moment theory, for convergence results on the DPS hierarchy from (A.C. Doherty, P.A. Parrilo and F.M. Spedalieri. Distinguishing separable and entangled states. Phys. Rev. Lett. 88(18):187904, 2002).

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arXiv2021

Exactness of Parrilo's conic approximations for copositive matrices and associated low order bounds for the stability number of a graph

De Klerk and Pasechnik (2002) introduced the bounds $\vartheta^{(r)}(G)$ ($r\in \mathbb{N}$) for the stability number $α(G)$ of a graph $G$ and conjectured exactness at order $α(G)-1$: $\vartheta^{(α(G)-1)}(G)=α(G)$. These bounds rely on the conic approximations $\mathcal{K}_n^{(r)}$ by Parrilo (2000) for the copositive cone $\text{COP}_n$. A difficulty in the convergence analysis of $\vartheta^{(r)}$ is the bad behaviour of the cones $\mathcal{K}_n^{(r)}$ under adding a zero row/column: when applied to a matrix not in $\mathcal{K}^{(0)}_n$ this gives a matrix not in any ${\mathcal{K}}^{(r)}_{n+1}$, thereby showing strict inclusion $\bigcup_{r\ge 0}{\mathcal{K}}^{(r)}_n\subset \text{COP}_n$ for $n\ge 6$. We investigate the graphs with $\vartheta^{(r)}(G)=α(G)$ for $r=0,1$: we algorithmically reduce testing exactness of $\vartheta^{(0)}$ to acritical graphs, we characterize critical graphs with $\vartheta^{(0)}$ exact, and we exhibit graphs for which exactness of $\vartheta^{(1)}$ is not preserved under adding an isolated node. This disproves a conjecture by Gvozdenović and Laurent (2007) which, if true, would have implied the above conjecture by de Klerk and Pasechnik.

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arXiv2020

Near-optimal analysis of Lasserre's univariate measure-based bounds for multivariate polynomial optimization

We consider a hierarchy of upper approximations for the minimization of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$ proposed recently by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the push-forward measure of the Lebesgue measure on $K$ by the polynomial $f$ and involves univariate sums of squares of polynomials with growing degrees $2r$. Hence it is weaker, but cheaper to compute, than an earlier hierarchy by Lasserre (SIAM Journal on Optimization 21(3), 864--885, 2011), which uses multivariate sums of squares. We show that this new hierarchy converges to the global minimum of $f$ at a rate in $O(\log^2 r / r^2)$ whenever $K$ satisfies a mild geometric condition, which holds, e.g., for convex bodies and for compact semialgebraic sets with dense interior. As an application this rate of convergence also applies to the stronger hierarchy based on multivariate sums of squares, which improves and extends earlier convergence results to a wider class of compact sets. Furthermore, we show that our analysis is near-optimal by proving a lower bound on the convergence rate in $Ω(1/r^2)$ for a class of polynomials on $K=[-1,1]$, obtained by exploiting a connection to orthogonal polynomials.

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arXiv2020

Optimizing hypergraph-based polynomials modeling job-occupancy in queueing with redundancy scheduling

We investigate two classes of multivariate polynomials with variables indexed by the edges of a uniform hypergraph and coefficients depending on certain patterns of union of edges. These polynomials arise naturally to model job-occupancy in some queuing problems with redundancy scheduling policy. The question, posed by Cardinaels, Borst and van Leeuwaarden (arXiv:2005.14566, 2020), is to decide whether their global minimum over the standard simplex is attained at the uniform probability distribution. By exploiting symmetry properties of these polynomials we can give a positive answer for the first class and partial results for the second one, where we in fact show a stronger convexity property of these polynomials over the simplex.

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arXiv2020

Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets

We consider the problem of computing the minimum value $f_{\min,K}$ of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$, which can be reformulated as finding a probability measure $ν$ on $K$ minimizing $\int_K f dν$. Lasserre showed that it suffices to consider such measures of the form $ν= qμ$, where $q$ is a sum-of-squares polynomial and $μ$ is a given Borel measure supported on $K$. By bounding the degree of $q$ by $2r$ one gets a converging hierarchy of upper bounds $f^{(r)}$ for $f_{\min,K}$. When $K$ is the hypercube $[-1, 1]^n$, equipped with the Chebyshev measure, the parameters $f^{(r)}$ are known to converge to $f_{\min,K}$ at a rate in $O(1/r^2)$. We extend this error estimate to a wider class of convex bodies, while also allowing for a broader class of reference measures, including the Lebesgue measure. Our analysis applies to simplices, balls and convex bodies that locally look like a ball. In addition, we show an error estimate in $O(\log r / r)$ when $K$ satisfies a minor geometrical condition, and in $O(\log^2 r / r^2)$ when $K$ is a convex body, equipped with the Lebesgue measure. This improves upon the currently best known error estimates in $O(1 / \sqrt{r})$ and $O(1/r)$ for these two respective cases.

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arXiv2019

Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere

We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure. We show that the exact rate of convergence is Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.

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arXiv2019

Semidefinite programming formulations for the completely bounded norm of a tensor

We show that a certain tensor norm, the completely bounded norm, can be expressed by a semidefinite program. This tensor norm recently attracted attention in the field of quantum computing, where it was used by Arunachalam, Briët and Palazuelos for characterizing the quantum query complexity of Boolean functions. Combined with their results, we obtain a new characterization of the quantum query complexity through semidefinite programming. Using the duality theory of semidefinite programming we obtain a new type of certificates for large query complexity. We show that our class of certificates encompasses the linear programming certificates corresponding to the approximate degree of a Boolean function.

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