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We consider the Moment-SOS hierarchy in polynomial optimization. We first provide a sufficient condition to solve the truncated K-moment problem associated with a given degree-$2n$ pseudo-moment sequence $\phi$ n and a semi-algebraic set $K \subset \mathbb{R}^d$. Namely, let $2v$ be the maximum degree of the polynomials that describe $K$. If the rank $r$ of its associated moment matrix is less than $nv + 1$, then $\phi^n$ has an atomic representing measure supported on at most $r$ points of $K$. When used at step-$n$ of the Moment-SOS hierarchy, it provides a sufficient condition to guarantee its finite convergence (i.e., the optimal value of the corresponding degree-n semidefinite relaxation of the hierarchy is the global minimum). For Quadratic Constrained Quadratic Problems (QCQPs) one may also recover global minimizers from the optimal pseudo-moment sequence. Our condition is in the spirit of Blekherman's rank condition and while on the one-hand it is more restrictive, on the other hand it applies to constrained POPs as it provides a localization on $K$ for the representing measure.

Christoffel polynomials are classical tools from approximation theory. They can be used to estimate the (compact) support of a measure $\mu$ on $\mathbb{R}^d$ based on its low-degree moments. Recently, they have been applied to problems in data science, including outlier detection and support inference. A major downside of Christoffel polynomials in such applications is the fact that, in order to compute their coefficients, one must invert a matrix whose size grows rapidly with the dimension $d$. In this paper, we propose a modification of the Christoffel polynomial which is significantly cheaper to compute, but retains many of its desirable properties. Our approach relies on sparsity of the underlying measure $\mu$, described by a graphical model. The complexity of our modification depends on the treewidth of this model., Comment: 21 pages, 1 figure

We introduce a variant of the D-optimal design of experiments problem with a more general information matrix that takes into account the representation of the design space S. The main motivation is that if S $\subset$ R d is the unit ball, the unit box or the canonical simplex, then remarkably, for every dimension d and every degree n, the equilibrium measure of S (in pluripotential theory) is an optimal solution. Equivalently, for each degree n, the unique optimal solution is the vector of moments (up to degree 2n) of the equilibrium measure of S. Hence nding an optimal design reduces to nding a cubature for the equilibrium measure, with atoms in S, positive weights, and exact up to degree 2n. In addition, any resulting sequence of atomic D-optimal measures converges to the equilibrium measure of S for the weak-star topology, as n increases. Links with Fekete sets of points are also discussed. More general compact basic semialgebraic sets are also considered, and a previously developed two-step design algorithm is easily adapted to this new variant of D-optimal design problem.

We introduce an infinite-dimensional version of the Christoffel function, where now (i) its argument lies in a Hilbert space of functions, and (ii) its associated underlying measure is supported on a compact subset of the Hilbert space. We show that it possesses the same crucial property as its finite-dimensional version to identify the support of the measure (and so to detect outliers). Indeed, the growth of its reciprocal with respect to its degree is at least exponential outside the support of the measure and at most polynomial inside. Moreover, for a fixed degree, its computation mimics that of the finite-dimensional case, but now the entries of the moment matrix associated with the measure are moments of moments. To illustrate the potential of this new tool, we consider the following application. Given a data base of registered reference trajectories, we consider the problem of detecting whether a newly acquired trajectory is abnormal (or out of distribution) with respect to the data base. As in the finite-dimensional case, we use the infinite-dimensional Christoffel function as a score function to detect outliers and abnormal trajectories. A few numerical examples are provided to illustrate the theory.

We consider a new multivariate generalization of the classical monic (univariate) Chebyshev polynomial that minimizes the uniform norm on the interval $[-1,1]$. Let $\Pi^*_n$ be the subset of polynomials of degree at most $n$ in $d$ variables, whose homogeneous part of degree $n$ has coefficients summing up to $1$. The problem is determining a polynomial in $\Pi^*_n$ with the smallest uniform norm on a domain $\Omega$, which we call a least Chebyshev polynomial (associated with $\Omega$). Our main result solves the problem for $\Omega$ belonging to a non-trivial class of domains that we call diagonally-determined, and establishes the remarkable result that a least Chebyshev polynomial can be given via the classical, univariate, Chebyshev polynomial. In particular, the solution can be independent of the dimension. Diagonally-determined domains include centered balls in $\mathbb{R}^d$ in any norm, but can be non-convex and highly irregular. We also introduce a computational procedure, based on semidefinite programming hierarchies, to detect if a given semi-algebraic set is diagonally-determined.

This paper explores methods for verifying the properties of Binary Neural Networks (BNNs), focusing on robustness against adversarial attacks. Despite their lower computational and memory needs, BNNs, like their full-precision counterparts, are also sensitive to input perturbations. Established methods for solving this problem are predominantly based on Satisfiability Modulo Theories and Mixed-Integer Linear Programming techniques, which are characterized by NP complexity and often face scalability issues. We introduce an alternative approach using Semidefinite Programming relaxations derived from sparse Polynomial Optimization. Our approach, compatible with continuous input space, not only mitigates numerical issues associated with floating-point calculations but also enhances verification scalability through the strategic use of tighter first-order semidefinite relaxations. We demonstrate the effectiveness of our method in verifying robustness against both $\|.\|_\infty$ and $\|.\|_2$-based adversarial attacks., Comment: 22 pages, 2 figures, 7 tables

This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a versatile semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all monomials up to degree six on the euclidean ball, the simplex, and the cross-polytope. Furthermore, inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before, namely for the monomial $x_1^2 x_2^2 x_3$ on the euclidean ball and for the monomial $x_1^2 x_2 x_3$ on the simplex.

Given a determinate (multivariate) probability measure $\mu$, we characterize Gaussian mixtures $\nu\_\phi$ which minimize the Wasserstein distance $W\_2(\mu,\nu\_\phi)$ to $\mu$ when the mixing probability measure $\phi$ on the parameters $(m,\Sigma)$ of the Gaussians is supported on a compact set $S$.(i) We first show that such mixtures are optimal solutions of a particular optimal transport (OT) problem where the marginal $\nu\_{\phi}$ of the OT problem is also unknown via the mixing measure variable $\phi$. Next (ii) by using a well-known specific property of Gaussian measures, this optimal transport is then viewed as a Generalized Moment Problem (GMP) and if the set $S$ of mixture parameters $(m,\Sigma)$ is a basic compact semi-algebraic set, we provide a "mesh-free" numerical scheme to approximate as closely as desired the optimal distance by solving a hierarchy of semidefinite relaxations of increasing size. In particular, we neither assume that the mixing measure is finitely supported nor that the variance is the same for all components. If the original measure $\mu$ is not a Gaussian mixture with parameters $(m,\Sigma)\in S$, then a strictly positive distance is detected at a finite step of the hierarchy. If the original measure $\mu$ is a Gaussian mixture with parameters $(m,\Sigma)\in S$, then all semidefinite relaxations of the hierarchy have same zero optimal value. Moreover if the mixing measure is atomic with finite support, its components can sometimes be extracted from an optimal solution at some semidefinite relaxation of the hierarchy when Curto & Fialkow's flatness condition holds for some moment matrix.

Given two measures $\mu$, $\nu$ on Rd that satisfy Carleman's condition, we provide a numerical scheme to approximate as closely as desired the total variation distance between $\mu$ and $\nu$. It consists of solving a sequence (hierarchy) of convex relaxations whose associated sequence of optimal values converges to the total variation distance, an additional illustration of the versatility of the Moment-SOS hierarchy. Indeed each relaxation in the hierarchy is a semidefinite program whose size increases with the number of involved moments. It has an optimal solution which is a couple of degree-2n pseudo-moments which converge, as n grows, to moments of the Hahn-Jordan decomposition of $\mu$-$\nu$.

Set-membership estimation (SME) outputs a set estimator that guarantees to cover the groundtruth. Such sets are, however, defined by (many) abstract (and potentially nonconvex) constraints and therefore difficult to manipulate. We present tractable algorithms to compute simple and tight overapproximations of SME in the form of minimum enclosing ellipsoids (MEE). We first introduce the hierarchy of enclosing ellipsoids proposed by Nie and Demmel (2005), based on sums-of-squares relaxations, that asymptotically converge to the MEE of a basic semialgebraic set. This framework, however, struggles in modern control and perception problems due to computational challenges. We contribute three computational enhancements to make this framework practical, namely constraints pruning, generalized relaxed Chebyshev center, and handling non-Euclidean geometry. We showcase numerical examples on system identification and object pose estimation., Comment: Accepted to 6th Learning for Dynamics and Control (L4DC) as oral presentation

We extend the polynomial Pell's equation satisfied by univariate Chebyshev polynomials on [--1, 1] from one variable to several variables, using orthogonal polynomials on regular domains that include cubes, balls, and simplexes of arbitrary dimension. Moreover, we show that such an equation is strongly connected (i) to a certificate of positivity (from real algebraic geometry) on the domain, as well as (ii) to the Christoffel functions of the equilibrium measure on the domain. In addition, the solution to Pell's equation reflects an extremal property of orthonormal polynomials associated with an entropy-like criterion., Comment: To appear in Transactions of the Amercian Mathematical Society

AC-OPF (Alternative Current Optimal Power Flow)aims at minimizing the operating costs of a power gridunder physical constraints on voltages and power injections.Its mathematical formulation results in a nonconvex polynomial optimizationproblem which is hard to solve in general,but that can be tackled by a sequence of SDP(Semidefinite Programming) relaxationscorresponding to the steps of the moment-SOS (Sums-Of-Squares) hierarchy.Unfortunately, the size of these SDPs grows drastically in the hierarchy,so that even second-order relaxationsexploiting the correlative sparsity pattern of AC-OPFare hardly numerically tractable for largeinstances -- with thousands of power buses.Our contribution lies in a new sparsityframework, termed minimal sparsity, inspiredfrom the specific structure of power flowequations.Despite its heuristic nature, numerical examples show that minimal sparsity allows the computation ofhighly accurate second-order moment-SOS relaxationsof AC-OPF, while requiring far less computing time and memory resources than the standard correlative sparsity pattern. Thus, we manage to compute second-order relaxations on test caseswith about 6000 power buses, which we believe to be unprecedented.

We briefly recall basics of the Moment-SOS hierarchy in polynomial optimization and the Christoffel-Darboux kernel (and the Christoffel function (CF)) in theory of approximation and orthogonal polynomials. We then (i) show a strong link between the CF and the SOS-based positive certificate at the core of the Moment-SOS hierarchy, and (ii) describe how the CD-kernel provides a simple interpretation of the SOS-hierarchy of lower bounds as searching for some signed polynomial density (while the SOS-hierarchy of upper bounds is searching for a positive (SOS) density). This link between the CF and positive certificates, in turn allows us (i) to establish a disintegration property of the CF much like for measures, and (ii) for certain sets, to relate the CF of their equilibrium measure with a certificate of positivity on the set, for constant polynomials.

We show that Bernstein polynomials are related to the Lebesgue measure on [0, 1] in a manner similar as Chebyshev polynomials are related to the equilibrium measure of [--1, 1]. We also show that Pell's polynomial equation satisfied by Chebyshev polynomials, provides a partition of unity of [--1, 1], the analogue of the partition of unity of [0, 1] provided by Bernstein polynomials. Both partitions of unity are interpreted as a specific algebraic certificate that the constant polynomial ''1'' is positive-on [--1, 1] via Putinar's certificate of positivity (for Chebyshev), and-on [0, 1] via Handeman's certificate of positivity (for Bernstein). Then in a second step, one combines this partition of unity with an interpretation of a duality result of Nesterov in convex conic optimization to obtain an explicit connection with the equilibrium measure on [--1, 1] (for Chebyshev) and Lebesgue measure on [0, 1] (for Bernstein). Finally this connection is also partially established for the ''d''-dimensional simplex.

We propose to approximate a (possibly discontinuous) multivariate function f (x) on a compact set by the partial minimizer arg miny p(x, y) of an appropriate polynomial p whose construction can be cast in a univariate sum of squares (SOS) framework, resulting in a highly structured convex semidefinite program. In a number of non-trivial cases (e.g. when f is a piecewise polynomial) we prove that the approximation is exact with a low-degree polynomial p. Our approach has three distinguishing features: (i) It is mesh-free and does not require the knowledge of the discontinuity locations. (ii) It is model-free in the sense that we only assume that the function to be approximated is available through samples (point evaluations). (iii) The size of the semidefinite program is independent of the ambient dimension and depends linearly on the number of samples. We also analyze the sample complexity of the approach, proving a generalization error bound in a probabilistic setting. This allows for a comparison with machine learning approaches.

We introduce a certain variant (or regularization) $\tilde{\Lambda}^\mu_n$ of the standard Christoffel function $\Lambda^\mu_n$ associated with a measure $\mu$ on a compact set $\Omega\subset \mathbb{R}^d$. Its reciprocal is now a sum-of-squares polynomial in the variables $(x,\varepsilon)$, $\varepsilon>0$. It shares the same dichotomy property of the standard Christoffel function, that is, the growth with $n$ of its inverse is at most polynomial inside and exponential outside the support of the measure. Its distinguishing and crucial feature states that for fixed $\varepsilon>0$, and under weak assumptions, $\lim_{n\to\infty} \varepsilon^{-d}\tilde{\Lambda}^\mu_n(\xi,\varepsilon)=f(\zeta_\varepsilon)$ where $f$ (assumed to be continuous) is the unknown density of $\mu$ w.r.t. Lebesgue measure on $\Omega$, and $\zeta_\varepsilon\in\mathbf{B}_\infty(\xi,\varepsilon)$ (and so $f(\zeta_\varepsilon)\approx f(\xi)$ when $\varepsilon>0$ is small). This is in contrast with the standard Christoffel function where if $\lim_{n\to\infty} n^d\Lambda^\mu_n(\xi)$ exists, it is of the form $f(\xi)/\omega_E(\xi)$ where $\omega_E$ is the density of the equilibrium measure of $\Omega$, usually unknown. At last but not least, the additional computational burden (when compared to computing $\Lambda^\mu_n$) is just integrating symbolically the monomial basis $(x^{\alpha})_{\alpha\in\mathbb{N}^d_n}$ on the box $\{x: \Vert x-\xi\Vert_\infty<\varepsilon/2\}$, so that $1/\tilde{\Lambda}^\mu_n$ is obtained as an explicit polynomial of $(\xi,\varepsilon)$., Comment: Rapport LAAS n 23003

We first interpret Pell's equation satisfied by Chebyshev polynomials for each degree t, as a certain Positivstellensatz, which then yields for each integer t, what we call a generalized Pell's equation, satisfied by reciprocals of Christoffel functions of ''degree'' 2t, associated with the equilibrium measure $\mu$ of the interval [--1, 1] and the measure (1 -- x 2)d$\mu$. We next extend this point of view to arbitrary compact basic semi-algebraic set S $\subset$ R n and obtain a generalized Pell's equation (by analogy with the interval [--1, 1]). Under some conditions, for each t the equation is satisfied by reciprocals of Christoffel functions of ''degree'' 2t associated with (i) the equilibrium measure $\mu$ of S and (ii), measures gd$\mu$ for an appropriate set of generators g of S. These equations depend on the particular choice of generators that define the set S. In addition to the interval [--1, 1], we show that for t = 1, 2, 3, the equations are indeed also satisfied for the equilibrium measures of the 2D-simplex, the 2D-Euclidean unit ball and unit box. Interestingly, this view point connects orthogonal polynomials, Christoffel functions and equilibrium measures on one side, with sum-of-squares, convex optimization and certificates of positivity in real algebraic geometry on another side.

We consider polynomial optimization problems (POP) on a semialgebraic set contained in the nonnegative orthant (every POP on a compact set can be put in this format by a simple translation of the origin). Such a POP can be converted to an equivalent POP by squaring each variable. Using even symmetry and the concept of factor width, we propose a hierarchy of semidefinite relaxations based on the extension of P\'olya's Positivstellensatz by Dickinson-Povh. As its distinguishing and crucial feature, the maximal matrix size of each resulting semidefinite relaxation can be chosen arbitrarily and in addition, we prove that the sequence of values returned by the new hierarchy converges to the optimal value of the original POP at the rate $O(\varepsilon^{-c})$ if the semialgebraic set has nonempty interior. When applied to (i) robustness certification of multi-layer neural networks and (ii) computation of positive maximal singular values, our method based on P\'olya's Positivstellensatz provides better bounds and runs several hundred times faster than the standard Moment-SOS hierarchy., Comment: 39 pages, 15 tables

A classical result from topology called Uryshon's lemma asserts the existence of a continuous separator of two disjoint closed sets in a sufficiently regular topological space. In this work we make a search for this separator constructive and efficient in the context of real algebraic geometry. Namely, given two compact disjoint basic semialgebraic sets which are contained in an $n$-dimensional box, we provide an algorithm that computes a separating polynomial greater than or equal to 1 on the first set and less than or equal to 0 on the second one., Comment: 4 pages, 1 figure, submitted as en extended abstract for the last POEMA workshop

Let $\mathcal{C}_{d,n}$ be the convex cone consisting of real $n$-variate degree $d$ forms that are strictly positive on $\mathbb{R}^n\setminus \{\mathbf{0}\}$. We prove that the Lebesgue volume of the sublevel set $\{g\leq 1\}$ of $g\in \mathcal{C}_{d,n}$ is a completely monotone function on $\mathcal{C}_{d,n}$ and investigate the related properties. Furthermore, we provide (partial) characterization of forms, whose sublevel sets have finite Lebesgue volume. Finally, we discover an interesting property of a centered Gaussian distribution, establishing a connection between the matrix of its degree $d$ moments and the quadratic form given by the inverse of its covariance matrix., Comment: 21 pages

We consider polynomials of a few linear forms and show how exploit this type of sparsity for optimization on some particular domains like the Euclidean sphere or a polytope. Moreover, a simple procedure allows to detect this form of sparsity and also allows to provide an approximation of any polynomial by such sparse polynomials.

We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of "the CF of the disintegration is the disintegration of the CFs". In the proof one uses an apparently overlooked property (but interesting in its own) which states that any sum-of-squares polynomial is the Christoffel function of some linear form (with a representing measure in the univariate case). The same is true for the convex cone of polynomials that are positive on a basic semi-algebraic set. This interpretation of the CF establishes another bridge between polynomials optimization and orthogonal polynomials.

We show that the empirical Christoffel function associated with a cloud of finitely many points sampled from a distribution, can provide a simple tool for supervised classification in data analysis, with good generalization properties.

In this paper we consider the problem of finding bounds on the prices of options depending on multiple assets without assuming any underlying model on the price dynamics, but only the absence of arbitrage opportunities. We formulate this as a generalized moment problem and utilize the well-known Moment-Sum-of-Squares (SOS) hierarchy of Lasserre to obtain bounds on the range of the possible prices. A complementary approach (also due to Lasserre) is employed for comparison. We present several numerical examples to demonstrate the viability of our approach. The framework we consider makes it possible to incorporate different kinds of observable data, such as moment information, as well as observable prices of options on the assets of interest., Comment: 20 pages

We report the experimental results on certifying 1% global optimality of solutions of AC-OPF instances from PGLiB via the CS-TSSOS hierarchy -- a moment-SOS based hierarchy that exploits both correlative and term sparsity, which can provide tighter SDP relaxations than Shor's relaxation. Our numerical experiments demonstrate that the CS-TSSOS hierarchy scales well with the problem size and is indeed useful in certifying global optimality of solutions for large-scale real world problems, e.g., the AC-OPF problem. In particular, we are able to certify 1% global optimality for a challenging AC-OPF instance with 6515 buses involving 14398 real variables and 63577 constraints., Comment: 18 pages, 5 figures, 4 tables

Deep equilibrium models are based on implicitly defined functional relations and have shown competitive performance compared with the traditional deep networks. Monotone operator equilibrium networks (monDEQ) retain interesting performance with additional theoretical guaranties. Existing certification tools for classical deep networks cannot directly be applied to monDEQs for which much fewer tools exist. We introduce a semialgebraic representation for ReLU based monDEQs which allows to approximate the corresponding input output relation by semidefinite programming (SDP). We present several applications to network certification and obtain SDP models for the following problems : robustness certification, Lipschitz constant estimation, ellipsoidal uncertainty propagation. We use these models to certify robustness of monDEQs w.r.t. a general $L_q$ norm. Experimental results show that the proposed models outperform existing approaches for monDEQ certification. Furthermore, our investigations suggest that monDEQs are much more robust to $L_2$ perturbations than $L_{\infty}$ perturbations., Comment: 16 pages, 4 tables, 2 figures

We introduce a sublevel Moment-SOS hierarchy where each SDP relaxation can be viewed as an intermediate (or interpolation) between the d-th and (d+1)-th order SDP relaxations of the Moment-SOS hierarchy (dense or sparse version). With the flexible choice of determining the size (level) and number (depth) of subsets in the SDP relaxation, one is able to obtain different improvements compared to the d-th order relaxation, based on the machine memory capacity. In particular, we provide numerical experiments for d=1 and various types of problems both in combinatorial optimization (Max-Cut, Mixed Integer Programming) and deep learning (robustness certification, Lipschitz constant of neural networks), where the standard Lasserre's relaxation (or its sparse variant) is computationally intractable. In our numerical results, the lower bounds from the sublevel relaxations improve the bound from Shor's relaxation (first order Lasserre's relaxation) and are significantly closer to the optimal value or to the best-known lower/upper bounds., Comment: 25 pages, 13 tables

We prove that every semidefinite moment relaxation of a polynomial optimization problem (POP) with a ball constraint can be reformulated as a semidefinite program involving a matrix with constant trace property (CTP). As a result such moment relaxations can be solved efficiently by first-order methods that exploit CTP, e.g., the conditional gradient-based augmented Lagrangian method. We also extend this CTP-exploiting framework to large-scale POPs with different sparsity structures. The efficiency and scalability of our framework are illustrated on second-order moment relaxations for various randomly generated quadratically constrained quadratic programs., Comment: 43 pages, 6 algorithms, 23 tables

This paper is concerned with minimizing a sum of rational functions over a compact set of high-dimension. Our approach relies on the second Lasserre's hierarchy (also known as the upper bounds hierarchy) formulated on the pushforward measure in order to work in a space of smaller dimension. We show that in the general case the minimum can be approximated as closely as desired from above with a hierarchy of semidefinite programs problems or, in the particular case of a single fraction, with a hierarchy of generalized eigenvalue problems. We numerically illustrate the potential of using the pushforward measure rather than the standard upper bounds hierarchy. In our opinion, this potential should be a strong incentive to investigate a related challenging problem interesting in its own; namely integrating an arbitrary power of a given polynomial on a simple set (e.g., unit box or unit sphere) with respect to Lebesgue or Haar measure., Comment: 24 pages, 3 tables

We consider the global minimization of a polynomial on a compact set B. We show that each step of the Moment-SOS hierarchy has a nice and simple interpretation that complements the usual one. Namely, it computes coefficients of a polynomial in an orthonormal basis of L 2 (B, $\mu$) where $\mu$ is an arbitrary reference measure whose support is exactly B. The resulting polynomial is a certain density (with respect to $\mu$) of some signed measure on B. When some relaxation is exact (which generically takes place) the coefficients of the optimal polynomial density are values of orthonormal polynomials at the global minimizer and the optimal (signed) density is simply related to the Christoffel-Darboux (CD) kernel and the Christoffel function associated with $\mu$. In contrast to the hierarchy of upper bounds which computes positive densities, the global optimum can be achieved exactly as integration against a polynomial (signed) density because the CD-kernel is a reproducing kernel, and so can mimic a Dirac measure (as long as finitely many moments are concerned).

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

We propose a novel method for planning shortest length piecewise-linear motions through complex environments punctured with static, moving, or even morphing obstacles. Using a moment optimization approach, we formulate a hierarchy of semidefinite programs that yield increasingly refined lower bounds converging monotonically to the optimal path length. For computational tractability, our global moment optimization approach motivates an iterative motion planner that outperforms competing sampling-based and nonlinear optimization baselines. Our method natively handles continuous time constraints without any need for time discretization, and has the potential to scale better with dimensions compared to popular sampling-based methods.

We consider degree-d forms on the Euclidean unit sphere. We specialize to our setting a genericity result by Nie obtained in a more general framework. We exhibit an homogeneous polynomial Res in the coefficients of f , such that if Res(f) = 0 then all points that satisfy first-and second-order necessary optimality conditions are in fact local minima of f on the unit sphere. Then we obtain obtain a simple and compact characterization of all local minima of generic degree-d forms, solely in terms of the value of (i) f , (ii) the norm of its gradient, and (iii) the first two smallest eigenvalues of its Hessian, all evaluated at the point. In fact this property also holds for twice continuous differentiable functions that are positively homogeneous. Finally we obtain a characterization of generic degree-d forms with no spurious local minimum on the unit sphere by using a property of gradient ideals in algebraic geometry.

We consider the problem of computing the Lebesgue volume of compact basic semi-algebraic sets. In full generality, it can be approximated as closely as desired by a converging hierarchy of upper bounds obtained by applying the Moment-SOS (sums of squares) methodology to a certain infinite-dimensional linear program (LP). At each step one solves a semidefinite relaxation of the LP which involves pseudo-moments up to a certain degree. Its dual computes a polynomial of same degree which approximates from above the discontinuous indicator function of the set, hence with a typical Gibbs phenomenon which results in a slow convergence of the associated numerical scheme. Drastic improvements have been observed by introducing in the initial LP additional linear moment constraints obtained from a certain application of Stokes' theorem for integration on the set. However and so far there was no rationale to explain this behavior. We provide a refined version of this extended LP formulation. When the set is the smooth super-level set of a single polynomial, we show that the dual of this refined LP has an optimal solution which is a continuous function.Therefore in this dual one now approximates a continuous function by a polynomial, hence with no Gibbs phenomenon, which explains and improves the already observed drastic acceleration of the convergence of the hierarchy. Interestingly, the technique of proof involves recent results on Poisson's partial differential equation (PDE).

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.

We show that (i) any constrained polynomial optimization problem (POP) has an equivalent formulation on a variety contained in an Euclidean sphere and (ii) the resulting semidefinite relaxations in the moment-SOS hierarchy have the constant trace property (CTP) for the involved matrices. We then exploit the CTP to avoid solving the semidefinite relaxations via interior-point methods and rather use ad-hoc spectral methods that minimize the largest eigenvalue of a matrix pencil. Convergence to the optimal value of the semidefinite relaxation is guaranteed. As a result we obtain a hierarchy of nonsmooth "spectral relaxations" of the initial POP. Efficiency and robustness of this spectral hierarchy is tested against several equality constrained POPs on a sphere as well as on a sample of randomly generated quadratically constrained quadratic problems (QCQPs)., Comment: 38 pages, 6 figures, 11 tables

This work proposes a new moment-SOS hierarchy, called CS-TSSOS, for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two existing frameworks for sparse polynomial optimization. The former is due to Waki et al. while the latter was initially proposed by Wang et al. and later exploited in the TSSOS hierarchy. In doing so we obtain CS-TSSOS -- a two-level hierarchy of semidefinite programming relaxations with (i), the crucial property to involve blocks of SDP matrices and (ii), the guarantee of convergence to the global optimum under certain conditions. We demonstrate its efficiency and scalability on several large-scale instances of the celebrated Max-Cut problem and the important industrial optimal power flow problem, involving up to six thousand variables and tens of thousands of constraints., Comment: 28 pages, 8 figures, 8 tables

This work is a follow-up and a complement to arXiv:1912.08899 [math.OC] for solving polynomial optimization problems (POPs). The chordal-TSSOS hierarchy that we propose is a new sparse moment-SOS framework based on term-sparsity and chordal extension. By exploiting term-sparsity of the input polynomials we obtain a two-level hierarchy of semidefinite programming relaxations. The novelty and distinguishing feature of such relaxations is to obtain quasi block-diagonal matrices obtained in an iterative procedure that performs chordal extension of certain adjacency graphs. The graphs are related to the terms arising in the original data and not to the links between variables. Various numerical examples demonstrate the efficiency and the scalability of this new hierarchy for both unconstrained and constrained POPs. The two hierarchies are complementary. While the former TSSOS arXiv:1912.08899 [math.OC] has a theoretical convergence guarantee, the chordal-TSSOS has superior performance but lacks this theoretical guarantee., Comment: 29 pages, 14 tables, 5 figures. arXiv admin note: text overlap with arXiv:1912.08899

If $f$ is a positive definite form, Reznick's Positivstellensatz [Mathematische Zeitschrift. 220 (1995), pp. 75--97] states that there exists $k\in\mathbf{N}$ such that ${\| x \|^{2k}_2}f$ is a sum of squares of polynomials. Assuming that $f$ can be written as a sum of forms $\sum_{l=1}^p f_l$, where each $f_l$ depends on a subset of the initial variables, and assuming that these subsets satisfy the so-called running intersection property, we provide a sparse version of Reznick's Positivstellensatz. Namely, there exists $k \in \mathbf{N}$ such that $f=\sum_{l = 1}^p {{\sigma_l}/{H_l^{k}}}$, where $\sigma_l$ is a sum of squares of polynomials, $H_l$ is a uniform polynomial denominator, and both polynomials $\sigma_l,H_l$ involve the same variables as $f_l$, for each $l=1,\dots,p$. In other words, the sparsity pattern of $f$ is also reflected in this sparse version of Reznick's certificate of positivity. We next use this result to also obtain positivity certificates for (i) polynomials nonnegative on the whole space and (ii) polynomials nonnegative on a (possibly non-compact) basic semialgebraic set, assuming that the input data satisfy the running intersection property. Both are sparse versions of a positivity certificate due to Putinar and Vasilescu., Comment: 19 pages, 2 tables

The Lipschitz constant of a network plays an important role in many applications of deep learning, such as robustness certification and Wasserstein Generative Adversarial Network. We introduce a semidefinite programming hierarchy to estimate the global and local Lipschitz constant of a multiple layer deep neural network. The novelty is to combine a polynomial lifting for ReLU functions derivatives with a weak generalization of Putinar's positivity certificate. This idea could also apply to other, nearly sparse, polynomial optimization problems in machine learning. We empirically demonstrate that our method provides a trade-off with respect to state of the art linear programming approach, and in some cases we obtain better bounds in less time., Comment: NeurIPS 2020