arXiv subjects

Jean Lasserre

Publications and source records attributed to Jean Lasserre.

At least 19 recordsLinked to original sources

arXiv2021

Graph Recovery From Incomplete Moment Information

We investigate a class of moment problems, namely recovering a measure supported on the graph of a function from partial knowledge of its moments, as for instance in some problems of optimal transport or density estimation. We show that the sole knowledge of first degree moments of the function, namely linear measurements, is sufficient to obtain asymptotically all the other moments by solving a hierarchy of semidefinite relax-ations (viewed as moment matrix completion problems) with a specific sparsity inducing criterion related to a weighted 1-norm of the moment sequence of the measure. The resulting sequence of optimal solutions converges to the whole moment sequence of the measure which is shown to be the unique optimal solution of a certain infinite-dimensional linear optimization problem (LP). Then one may recover the function by a recent extraction algorithm based on the Christoffel-Darboux kernel associated with the measure. Finally, the support of such a measure supported on a graph is a meager, very thin (hence sparse) set. Therefore the LP on measures with this sparsity inducing criterion can be interpreted as an analogue for infinite-dimensional signals of the LP in super-resolution for (sparse) atomic signals. In data science, it is often relevant to process moments of a signal instead of the signal itself. For complex valued signals, the moments are Fourier coefficients, and many filtering operations are efficiently carried out in the sequence of moments. In numerical approximation algorithms, many operations on real valued signals are more efficiently implemented in their sequence of Chebyshev coefficients [19]. In the moment-SOS hierarchy approach, many nonlinear nonconvex problems are reformulated and solved approximately in the sequence of moments; see [13, 12] and references therein. Once moments or approximate moments have been computed, one is faced with the inverse problem of reconstructing the signal from its moments. The recent work [15] describes an algorithm based on the Christoffel-Darboux kernel, to recover the graph of a function from knowledge of its moments. Its novelty (and distinguishing feature) is to approximate the graph of the function (rather than the function itself) with a semialgebraic function (namely a minimizer of a sum of squares of polynomials) with L 1 and pointwise convergence guarantees for an increasing number of input moments. In

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arXiv2021

Semi-algebraic approximation using Christoffel-Darboux kernel

We provide a new method to approximate a (possibly discontinuous) function using Christoffel-Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the graph of the function. Such an input is available when approximating weak (or measure-valued) solution of optimal control problems, entropy solutions to non-linear hyperbolic PDEs, or using numerical integration from finitely many evaluations of the function. While most of the existing methods construct a piecewise polynomial approximation, we construct a semi-algebraic approximation whose estimation and evaluation can be performed efficiently. An appealing feature of this method is that it deals with non-smoothness implicitly so that a single scheme can be used to treat smooth or non-smooth functions without any prior knowledge. On the theoretical side, we prove pointwise convergence almost everywhere as well as convergence in the Lebesgue one norm under broad assumptions. Using more restrictive assumptions, we obtain explicit convergence rates. We illustrate our approach on various examples from control and approximation. In particular we observe empirically that our method does not suffer from the the Gibbs phenomenon when approximating discontinuous functions.

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arXiv2020

Dual optimal design and the Christoffel-Darboux polynomial

The purpose of this short note is to show that the Christoffel-Darboux polynomial, useful in approximation theory and data science, arises naturally when deriving the dual to the problem of semi-algebraic D-optimal experimental design in statistics. It uses only elementary notions of convex analysis. Geometric interpretations and algorithmic consequences are mentioned.

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arXiv2020

Simple Formula for Integration of Polynomials on a Simplex

We show that integrating a polynomial of degree t on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating t homogeneous polynomials of degree j = 1, 2,. .. , t, each at a unique point $ξ$ j of the simplex. This new and very simple formula can be exploited in finite (and extended finite) element methods, as well as in other applications where such integrals are needed.

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arXiv2020

Connecting optimization with spectral analysis of tri-diagonal matrices

We show that the global minimum (resp. maximum) of a continuous function on a compact set can be approximated from above (resp. from below) by computing the smallest (rest. largest) eigenvalue of a hierarchy of (r x r) tri-diagonal univariate moment matrices of increasing size. Equivalently it reduces to computing the smallest (resp. largest) root of a certain univariate degree-r orthonormal polynomial. This provides a strong connection between the fields of optimization, orthogonal polynomials, numerical analysis and linear algebra, via asymptotic spectral analysis of tri-diagonal symmetric matrices.

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arXiv2020

Approximating regions of attraction of a sparse polynomial differential system *

Motivated by stability analysis of large scale power systems, we describe how the Lasserre (moment-sums of squares, SOS) hierarchy can be used to generate outer approximations of the region of attraction (ROA) of sparse polynomial differential systems, at the price of solving linear matrix inequalities (LMI) of increasing size. We identify specific sparsity structures for which we can provide numerically certified outer approximations of the region of attraction in high dimension. For this purpose, we combine previous results on non-sparse ROA approximations with sparse semi-algebraic set volume computation.

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arXiv2018

Distributionally robust polynomial chance-constraints under mixture ambiguity sets

Given $X \subset R^n$, $\varepsilon \in (0,1)$, a parametrized family of probability distributions $(μ\_{a})\_{a\in A}$ on $Ω\subset R^p$, we consider the feasible set $X^*\_\varepsilon\subset X$ associated with the {\em distributionally robust} chance-constraint \[X^*\_\varepsilon\,=\,\{x \in X :\:{\rm Prob}\_μ[f(x,ω)\,>\,0]> 1-\varepsilon,\,\forallμ\in M\_a\},\]where $M\_a$ is the set of all possibles mixtures of distributions $μ\_a$, $a\in A$.For instance and typically, the family$M\_a$ is the set of all mixtures ofGaussian distributions on $R$ with mean and standard deviation $a=(a,σ)$ in some compact set $A\subset R^2$.We provide a sequence of inner approximations $X^d\_\varepsilon=\{x\in X: w\_d(x) <\varepsilon\}$, $d\in N$, where $w\_d$ is a polynomial of degree $d$ whosevector of coefficients is an optimal solution of a semidefinite program.The size of the latter increases with the degree $d$. We also obtain the strong and highly desirable asymptotic guarantee that $λ(X^*\_\varepsilon\setminus X^d\_\varepsilon)\to0$as $d$ increases, where $λ$ is the Lebesgue measure on $X$. Same resultsare also obtained for the more intricated case of distributionally robust "joint" chance-constraints.

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arXiv2018

Volume Of Sub-level Sets Of Homogeneous Polynomials

Consider the sub level set K := {x : g(x) $\le$ 1} where g is a positive and homogeneous polynomial. We show that its Lebesgue volume can be approximated as closely as desired by solving a sequence of generalized eigenvalue problems with respect to a pair of Hankel matrices of increasing size, and whose entries are obtained in closed form.

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arXiv2018

The Moment-SOS hierarchy

The Moment-SOS hierarchy initially introduced in optimization in 2000, is based on the theory of the K-moment problem and its dual counterpart, polynomials that are positive on K. It turns out that this methodology can be also applied to solve problems with positivity constraints " f (x) $\ge$ 0 for all x $\in$ K " and/or linear constraints on Borel measures. Such problems can be viewed as specific instances of the " Generalized Problem of Moments " (GPM) whose list of important applications in various domains is endless. We describe this methodology and outline some of its applications in various domains.

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arXiv2018

A moment approach for entropy solutions to nonlinear hyperbolic PDEs

We propose to solve polynomial hyperbolic partial differential equations (PDEs) with convex optimization. This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution, satisfying a linear equation in the space of Borel measures. The aim of this paper is, first, to provide the conditions that ensure the equivalence between the two formulations and, second, to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex, finite-dimensional, semidefinite programming problems. This result is then illustrated on the celebrated Burgers equation. We also compare our results with an existing numerical scheme, namely the Godunov scheme.

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arXiv2017

Lebesgue and gaussian measure of unions of basic semi-algebraic sets

Given a finite Borel measure $μ$ on R n and basic semi-algebraic sets $Ω$\_i $\subset$ R n , i = 1,. .. , p, we provide a systematic numerical scheme to approximate as closely as desired $μ$(\cup\_i $Ω$\_i), when all moments of $μ$ are available (and finite). More precisely , we provide a hierarchy of semidefinite programs whose associated sequence of optimal values is monotone and converges to the desired value from above. The same methodology applied to the complement R n \ (\cup\_i $Ω$\_i) provides a monotone sequence that converges to the desired value from below. When $μ$ is the Lebesgue measure we assume that $Ω$ := \cup\_i $Ω$\_i is compact and contained in a known box B and in this case the complement is taken to be B \ $Ω$. In fact, not only $μ$($Ω$) but also every finite vector of moments of $μ$\_$Ω$ (the restriction of $μ$ on $Ω$) can be approximated as closely as desired, and so permits to approximate the integral on $Ω$ of any given polynomial.

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arXiv2016

Bound-constrained polynomial optimization using only elementary calculations

We provide a monotone non increasing sequence of upper bounds $f^H_k$ ($k\ge 1$) converging to the global minimum of a polynomial $f$ on simple sets like the unit hypercube. The novelty with respect to the converging sequence of upper bounds in [J.B. Lasserre, A new look at nonnegativity on closed sets and polynomial optimization, SIAM J. Optim. 21, pp. 864--885, 2010] is that only elementary computations are required. For optimization over the hypercube, we show that the new bounds $f^H_k$ have a rate of convergence in $O(1/\sqrt {k})$. Moreover we show a stronger convergence rate in $O(1/k)$ for quadratic polynomials and more generally for polynomials having a rational minimizer in the hypercube. In comparison, evaluation of all rational grid points with denominator $k$ produces bounds with a rate of convergence in $O(1/k^2)$, but at the cost of $O(k^n)$ function evaluations, while the new bound $f^H_k$ needs only $O(n^k)$ elementary calculations.

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arXiv2012

A Lagrangian relaxation view of linear and semidefinite hierarchies

We consider the general polynomial optimization problem $P: f^*=\min \{f(x)\,:\,x\in K\}$ where $K$ is a compact basic semi-algebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the global optimum $f^*$, provided that it is applied to a problem $\tilde{P}$ equivalent to $P$, in which sufficiently many redundant constraints (products of the initial ones) are added to the initial description of $P$. Next we show that the standard hierarchy of LP-relaxations of $P$ (in the spirit of Sherali-Adams' RLT) can be interpreted as a brute force simplification of the above Lagrangian relaxation in which a nonnegative polynomial (with coefficients to be determined) is replaced with a constant polynomial equal to zero. Inspired by this interpretation, we provide a systematic improvement of the LP-hierarchy by doing a much less brutal simplification which results into a parametrized hierarchy of semidefinite programs (and not linear programs any more). For each semidefinite program in the parametrized hierarchy, the semidefinite constraint has a fixed size $O(n^k)$, independently of the rank in the hierarchy, in contrast with the standard hierarchy of semidefinite relaxations. The parameter $k$ is to be decided by the user. When applied to a non trivial class of convex problems, the first relaxation of the parametrized hierarchy is exact, in contrast with the LP-hierarchy where convergence cannot be finite. When applied to 0/1 programs it is at least as good as the first one in the hierarchy of semidefinite relaxations. However obstructions to exactness still exist and are briefly analyzed. Finally, the standard semidefinite hierarchy can also be viewed as a simplification of an extended Lagrangian relaxation, but different in spirit as sums of squares (and not scalars) multipliers are allowed.

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arXiv2012

Recovering an homogeneous polynomial from moments of its level set

Let $K:={x: g(x)\leq 1}$ be the compact sub-level set of some homogeneous polynomial $g$. Assume that the only knowledge about $K$ is the degree of $g$ as well as the moments of the Lebesgue measure on $K$ up to order 2d. Then the vector of coefficients of $g$ is solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order 2d of the Lebesgue measure on $K$ encode all information on the homogeneous polynomial $g$ that defines $K$ (in fact, only moments of order $d$ and 2d are needed).

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arXiv2012

The inverse moment problem for convex polytopes

The goal of this paper is to present a general and novel approach for the reconstruction of any convex d-dimensional polytope P, from knowledge of its moments. In particular, we show that the vertices of an N-vertex polytope in R^d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure od degree D) in d+1 distinct generic directions. Our approach is based on the collection of moment formulas due to Brion, Lawrence, Khovanskii-Pukhikov, and Barvinok that arise in the discrete geometry of polytopes, and what variously known as Prony's method, or Vandermonde factorization of finite rank Hankel matrices.

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arXiv2011

The K-moment problem for continuous linear functionals

Given a closed (and non necessarily compact) basic semi-algebraic set $K\subseteq R^n$, we solve the $K$-moment problem for continuous linear functionals. Namely, we introduce a weighted $\ell_1$-norm $\ell_w$ on $R[x]$, and show that the $\ell_w$-closures of the preordering $P$ and quadratic module $Q$ (associated with the generators of $K$) is the cone $psd(K)$ of polynomials nonnegative on $K$. We also prove that $P$ an $Q$ solve the $K$-moment problem for $\ell_w$-continuous linear functionals and completely characterize those $\ell_w$-continuous linear functionals nonnegative on $P$ and $Q$ (hence on $psd(K)$). When $K$ has a nonempty interior we also provide in explicit form a canonical $\ell_w$-projection $g^w_f$ for any polynomial $f$, on the (degree-truncated) preordering or quadratic module. Remarkably, the support of $g^w_f$ is very sparse and does not depend on $K$! This enables us to provide an explicit Positivstellensatz on $K$. At last but not least, we provide a simple characterization of polynomials nonnegative on $K$, which is crucial in proving the above results.

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arXiv2011

Existence of Gaussian cubature formulas

We provide a necessary and sufficient condition for existence of Gaussian cubature formulas. It consists of checking whether some overdetermined linear system has a solution and so complements Mysovskikh's theorem which requires computing common zeros of orthonormal polynomials. Moreover, the size of the linear system shows that existence of a cubature formula imposes severe restrictions on the associated linear functional. For fixed precision (or degree), the larger the number of variables the worse it gets. And for fixed number of variables, the larger the precision the worse it gets. Finally, we also provide an interpretation of the necessary and sufficient condition in terms of existence of a polynomial with very specific properties.

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arXiv2011

An algorithm for semi-infinite polynomial optimization

We consider the semi-infinite optimization problem: $f^*:=\min_{x\in X}\:\{f(x): g(x,y)\,\leq \,0,\:\forally\in Y_x\}$, where $f,g$ are polynomials and $X\subset R^n$ as well as $Y_\x\subset R^p$, $x\in X$, are compact basic semi-algebraic sets. To approximate $f^*$ we proceed in two steps. First, we use the "joint+marginal" approach of the author to approximate from above the function $x\mapstoΦ(x)=\sup \{g(x,y): y\in Y_x\}$ by a polynomial $Φ_d\geqΦ$, of degree at most $2d$, with the strong property that $Φ_d$ converges to $Φ$ for the $L_1$-norm, as $d\to\infty$ (and in particular, almost uniformly for some subsequence $(d_\ell)$, $\ell\in\N$). Then we solve the polynomial optimization problem $f^*_d=\min_{x\in X} \{f(x): Φ_d(x)\leq0\}$ via a (by now standard) hierarchy of semidefinite relaxations. It turns out that the optimal value $f^*_d\geq f^*$ converges to $f^*$ as $d\to\infty$. In practice we let $d$ be fixed, small, and relax the constraint $Φ_d\leq0$ to $Φ_d(x)\leqε$ with $ε>0$, allowing to change $ε$ dynamically.

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