arXiv subjects

Didier Henrion

Publications and source records attributed to Didier Henrion.

At least 19 recordsLinked to original sources

arXiv2026

Semidefinite relaxations for nonlinear elasticity with energies convex in the Cauchy-Green strain tensor

In nonlinear elasticity, finding the deformation of a material which minimizes a given stored energy density is a challenging calculus of variations problem which may fail to have minimizers: the energy optimal material forms infinitely fine microstructures (wrinkles) rather than deforming smoothly. In the case where the energy function is non-convex but frame indifferent and convex with respect to the Cauchy-Green strain tensor, we use the standard Le Dret-Raoult semidefinite projection formula for the quasiconvex envelope of the energy function together with a recent no gap result for convex calculus of variation problems to prove that there is no relaxation gap between the original non-convex calculus of variations problem and its linear moment formulation based on occupation measures. This implies convergence of the Lasserre moment-sum-of-squares (SOS) hierarchy and provides a computationally efficient, mesh-free numerical method that, unlike the finite element method, avoids undesirable mesh-dependent artifacts. Under the additional condition that the boundary condition is linear and the function is SOS convex in the strain tensor, we show that the first relaxation of the Lasserre hierarchy is exact. In other words, computing the quasiconvex envelope at a point boils down to solving a small convex semidefinite optimization problem.

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arXiv2026

Convergence rate of the moment-SOS hierarchy for univariate polynomial optimization

We study the convergence rate of the moment-SOS (sum-of-squares) hierarchy for polynomial optimization problems (POPs) on a bounded subset of the real line described by arbitrary polynomial inequalities. We prove that, for every fixed univariate POP, the relaxation error is bounded by $O(1/r^2)$, where $r$ is the relaxation order. In particular, boundary degeneracies in the polynomial description of the feasible set affect the constant but not the convergence exponent. The proof combines the structure of finitely generated univariate quadratic modules with a Chebyshev polynomial construction that approximately recovers the natural generators of the feasible set while controlling the degree of the certificate. We also give an elementary degree-four example for which the relaxation error is exactly $1/(2r(r-1))$, showing that the quadratic rate is optimal. Equivalent reformulations connect this example to a cubic univariate problem and to a bivariate POP whose feasible set has a cusp singularity at the minimizer.

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arXiv2026

Volume of quasi-homogeneous sublevel sets: Two linear algebra deterministic algorithms with convergence rates

We consider the problem of computing the Lebesgue volume of the unit sublevel set of a positive quasi-homogeneous polynomial. Pushing the Lebesgue measure of an ambient bounding box forward through the polynomial reduces this high-dimensional volume to a one-dimensional moment problem. This removes the ambient dimension from the optimization and confines the dimension to a single preprocessing stage, computing the moments of the polynomial over the box, which is polynomial in the ambient dimension for sparse or separable polynomials. We propose two deterministic algorithms for the resulting univariate relaxations, each returning certified upper and lower bounds on the volume. Both bypass semidefinite optimization entirely and rely only on standard numerical linear algebra. The first approximates a piecewise-constant function by a Chebyshev polynomial, so that each relaxation reduces to a fast cosine transform, and converges at a polynomial rate in the relaxation order. The second extracts the volume bounds from a single generalized eigenvalue problem involving moment and localizing matrices whose size grows linearly with the relaxation order, and converges at an exponential rate; the ratio governing this rate is determined by an a priori upper bound on the polynomial over the bounding box. Finally, the univariate polynomials produced by either algorithm are feasible for the multivariate moment-SOS volume hierarchy. The algebraic and geometric rates therefore transfer to the hierarchy itself, improving on its best known convergence rates.

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arXiv2026

Maximal entropy in the moment body

A moment body is a linear projection of the spectraplex, the convex set of trace-one positive semidefinite matrices. Determining whether a given point lies within a given moment body is a problem with numerous applications in quantum state estimation and polynomial optimization. This moment body membership oracle can be addressed with semidefinite programming, for which several off-the-shelf interior-point solvers are available. In this paper, inspired by techniques from quantum information theory, we argue analytically and geometrically that a much more efficient approach consists of minimizing globally a smooth strictly convex log-partition function, dual to a maximum entropy problem. We analyze the curvature properties of this function, showing that conditioning is governed by the distance of the point to the boundary of the moment body, and we describe a neat geometric preconditioning algorithm that exploits this analysis. Basic numerical experiments, comparing against interior-point and first-order semidefinite solvers, reveal a cubic dependence on the matrix size, similar to a few eigenstructure computations. They also illustrate the two regimes of the oracle: dense projections are handled efficiently up to sizes of several hundred, while sparse instances such as matrix completion scale to matrices of size several thousand in minutes on a standard laptop. In both cases the main bottleneck in this approach to large-scale semidefinite programming is moved almost entirely to efficient gradient storage and manipulation.

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arXiv2026

Polynomial-Based Solutions to Targeting Problems for Onboard Applications

This paper solves the targeting problem focusing on accuracy, computational efficiency, and reliability. The trajectory optimization problem is first recast as a polynomial optimization problem (POP) by leveraging differential algebra to compute high-order Taylor expansions of the nonlinear dynamics and constraints. Moment-sum-of-squares (SOS) optimization is then utilized to solve this POP. A convex formulation based on a second-order expansion of the dynamics is also proposed. For impulsive targeting, the moment-SOS and convex approaches are compared against traditional nonlinear programming (NLP) solvers and map inversion techniques. Results indicate that the moment-SOS approach provides solutions as accurate as traditional NLP, but with the critical advantage of guaranteeing convergence to the global optimum under mild assumptions. Furthermore, the method excels at handling large maneuvers and long propagation times, conditions in which standard linear approximations rapidly degrade. To demonstrate its versatility, the methodology is extended to a continuous low-thrust station keeping (SK) scenario in the Earth-Moon Circular Restricted Three-Body Problem. The algorithm's performance is then evaluated in the presence of significant state errors. The ability to directly handle non-convex constraints and recast complex, nonlinear dynamics into formulations with reliable convergence properties makes the moment-SOS approach suitable for autonomous onboard applications.

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arXiv2026

Extreme points and faces in the moment problem

The polyconvex envelope, used in the calculus of variations and elasticity theory, was expressed by Dacorogna pointwise as a linear program on finitely atomic measures on the space of $m\times n$ matrices. Weizsäcker and Winkler proved that the corresponding linear program on Borel measures restricts to the extreme points without increasing the infimum. Combining the two, one obtains a speed-up of grid-based algorithms and a new proof that the polyconvex envelope can be computed by the moment sum-of-squares hierarchy. Motivated by these applications, we seize the essence of extreme points in moment problems. First, we characterize extreme points of an affinely constrained convex set by the injectivity of the constraint map on the smallest faces containing them. We then study finitely many moment constraints. The extreme points are finitely atomic measures that have an affine independence property, under natural assumptions. We retrieve this known result with a simplified proof and apply it to faces of the probability simplex, among them the face of Radon measures. In the converse, we find that the assumption of a simplex is redundant. The Richter-Tchakaloff theorem allows us to show that the infimum of an integral functional restricts to the extreme points without increasing the infimum, not just for the known case of Radon measures but for any convex set of probability measures that contains the point measures.

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arXiv2026

Scalable anomaly detection via a univariate Christoffel function

Anomaly detection plays a critical role in identifying unusual patterns across domains such as fraud detection, network intrusion, and system fault diagnosis. Recently, Christoffel function-based methods, rooted in polynomial optimization, have emerged as promising alternatives to deep learning due to their strong mathematical foundations and computational frugality. However, their practical applicability is hindered by the need to invert a matrix whose size grows exponentially with the data dimension, rendering the method intractable even for moderate-dimensional datasets. This paper addresses the dimensionality limitations of Christoffel function-based anomaly detection while preserving its key theoretical properties, i.e., the on-off support dichotomy behavior and the accurate support shape capture. We introduce UCF, a univariate Christoffel function which is based on the squared distance between the query point and the support points. Extensive experiments on the ADBench benchmark demonstrate that UCF consistently outperforms 14 state-of-the-art baselines in terms of Average Precision. By resolving the scalability bottleneck of the Christoffel Function, this work expands the toolkit of anomaly detection methods with a robust, theoretically grounded, and universally applicable approach.

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arXiv2026

Concentration of measure-valued solutions for semilinear parabolic equations

The moment-sum-of-squares hierarchy provides a powerful framework for solving non-convex optimal control problems by constructing a sequence of convex semidefinite relaxations. However, when extending these methods to nonlinear partial differential equations (PDEs), a fundamental challenge is the potential existence of a relaxation gap, where the solution to the linear measure formulation using occupation measures fails to correspond to a classical physical solution of the original PDE. In this paper, we prove the absence of a relaxation gap for scalar semilinear parabolic PDEs of the reaction-diffusion type. We do so by showing that each solution to the linear measure equation gives rise to an energy measure-valued (emv) solution in the space of Young measures satisfying suitable energy identities. We then prove that any such emv solution concentrates on the solution to the nonlinear PDE, provided the latter exists and is unique. To the best of our knowledge, this is the first concentration result of this kind for measure-valued solutions of reaction-diffusion PDEs.

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arXiv2026

Composition and tensor train structure in polynomial optimization

We study polynomial optimization problems whose objective has a composition or tensor train structure. These polynomials can be evaluated as a sequence of maps, giving rise to intermediate variables (``states'') of dimension lower than the ambient dimension. Structures like these arise naturally in dynamical systems, Markov chains, and neural networks. We develop two moment-SOS (sums of squares) hierarchies that exploit this composition structure in different ways. The first one, termed state-lifting chordal, is based on the correlative sparsity graph of the problem. The second one, termed state-lifting push-forward, encodes the structure at the level of the measures directly. Numerical experiments demonstrate that the proposed methods can compute certified bounds for problems with hundreds or even a thousand variables. To illustrate the versatility of the hierarchies we apply them to Markov chain optimization, quantum optimal control, and neural networks.

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arXiv2026

Polyconvexity with Moments and Sums of Squares

A function of a matrix is polyconvex when it can be expressed as a convex function of the matrix minors. Polyconvexity is a regularity condition ensuring existence of minimizers in nonlinear elasticity and, more broadly, in vectorial problems of the calculus of variations, when minimizing integral gradient functionals. The polyconvex envelope of a function is the largest polyconvex lower bound. Yet deciding whether a given energy is polyconvex, or computing the polyconvex envelope, are generally difficult problems. This paper focuses on polynomial matrix functions. We propose (i) tractable convex-optimization based sufficient conditions to certify polyconvexity via sum-of-squares (SOS) technology, and (ii) a principled numerical method to compute the polyconvex envelope pointwise, based on the moment-SOS hierarchy from polynomial optimization.

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arXiv2026

Unsafe Probabilities and Risk Contours for Stochastic Processes using Convex Optimization

This paper proposes an algorithm to calculate the maximal probability of unsafety with respect to trajectories of a stochastic process and a hazard set. The unsafe probability estimation problem is cast as a primal-dual pair of infinite-dimensional linear programs in occupation measures and continuous functions. This convex relaxation is nonconservative (to the true probability of unsafety) under compactness and regularity conditions in dynamics. The continuous-function linear program is linked to existing probability-certifying barrier certificates of safety. Risk contours for initial conditions of the stochastic process may be generated by suitably modifying the objective of the continuous-function program, forming an interpretable and visual representation of stochastic safety for test initial conditions. All infinite-dimensional linear programs are truncated to finite dimension by the Moment-Sum-of-Squares hierarchy of semidefinite programs. Unsafe-probability estimation and risk contours are generated for example stochastic processes.

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arXiv2026

Mollified Christoffel-Darboux Kernels and Density Recovery on Varieties

We introduce mollified Christoffel-Darboux (CD) kernels on varieties, a systematic regularization of the classical CD kernel associated with a probability measure on a compact domain. The main motivations are twofold: first, to sharpen the classical on/off-support dichotomy of the CD polynomial by replacing linear growth on the support by a uniform bound; second, to obtain consistent and quantitatively controlled recovery of densities from moment data, without the need to know the equilibrium measure of the underlying domain. Our contributions are the following: (i) We introduce families of mollifiers on algebraic varieties. For each measure and degree on such a variety we define a mollified CD kernel, which can be computed from the moments of the underlying measure by linear algebra. (ii) We prove, by elementary arguments, that an improved dichotomy property holds: on the interior of the support the mollified CD polynomial is uniformly bounded in the degree, while outside the support it grows exponentially with the degree. (iii) Assuming Sobolev regularity of the density with respect to a reference measure, we derive explicit convergence rates for density recovery for measures in Euclidean space via mollified CD kernel. (iv) On the unit sphere, we show that suitably chosen algebraic mollifiers, constructed from zonal polynomials, lead to kernels with improved rates, building on classical constructive approximation results.

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arXiv2025

Solving Stengle's Example in Rational Arithmetic: Exact Values of the Moment-SOS Relaxations

We revisit Stengle's classical univariate polynomial optimization example $min 1 - x^2 s.t. (1 - x^2)^3 \geq 0$ whose constraint description is degenerate at the minimizers. We prove that the moment-SOS hierarchy of relaxation order $r \geq 3$ has the exact value $-1/r(r - 2)$. For this we construct in rational arithmetic a dual polynomial sum-of-squares (SOS) certificate and a primal moment sequence representing a finitely atomic measure. The key ingredients are elementary trigonometric properties of Chebyshev and Gegenbauer polynomial, and a Christoffel-Darboux kernel argument.

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arXiv2025

Global optimization of low-rank polynomials

This work considers polynomial optimization problems where the objective admits a low-rank canonical polyadic tensor decomposition. We introduce LRPOP (low-rank polynomial optimization), a new hierarchy of semidefinite programming relaxations for which the size of the semidefinite blocks is determined by the canonical polyadic rank rather than the number of variables. As a result, LRPOP can solve low-rank polynomial optimization problems that are far beyond the reach of existing sparse hierarchies. In particular, we solve problems with up to thousands of variables with total degree in the thousands. Numerical conditioning for problems of this size is improved by using the Bernstein basis. The LRPOP hierarchy converges from below to the global minimum of the polynomial under standard assumptions.

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arXiv2025

Density, Determinacy, Duality and a Regularized Moment-SOS Hierarchy

The standard moment-sum-of-squares (SOS) hierarchy is a powerful method for solving global polynomial optimization problems. However, its convergence relies on Putinar's Positivstellensatz, which requires the feasible set to satisfy the algebraic Archimedean property. In this paper, we introduce a regularized moment-SOS hierarchy capable of handling problems on unbounded sets or bounded sets violating the Archimedean property. Adopting a functional analysis viewpoint, we rely on the multivariate Carleman condition for measure determinacy rather than algebraic compactness. We prove that finite degree projections of the quadratic module are dense in the cone of positive polynomials with respect to the square norm induced by the measure. Based on these density results, we prove the convergence of a regularized hierarchy without invoking any Positivstellensatz. Furthermore, we propose a penalized formulation of the hierarchy which, combined with Bernstein-Markov inequalities, provides a monotonically non-decreasing sequence of certified lower bounds on the global minimum. The approach is illustrated on several benchmark problems known to be difficult or ill-posed for the standard hierarchy.

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arXiv2025

Polyconvex double well functions

We investigate polyconvexity of the double well function $f(X)\,:= |X-X\_1|^2|X-X\_2|^2$ for given matrices $X\_1, X\_2 \in \R^{n \times n}$. Such functions are fundamental in the modeling of phase transitions in materials, but their non-convex nature presents challenges for the analysis of variational problems. Polyconvexity of $f$ is related to the singular values of the matrix difference $X\_1 - X\_2$. We prove that $f$ is polyconvex if and only if the square of the largest singular value does not exceed the sum of the squares of the other singular values. This condition allows the function to be decomposed into the sum of a strictly convex part and a null Lagrangean. As a direct application of this result, we prove an existence and uniqueness theorem for the corresponding Dirichlet minimization problem of the integral functional.

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arXiv2025

Positively not SOS: pseudo-moments and extreme rays in exact arithmetic

A polynomial that is a sum of squares (SOS) of other polynomials is evidently positive. The converse is not true, there are positive polynomials which are not SOS. This note focuses on the problem of certifying, in exact arithmetic, that a given positive polynomial is not SOS. Using convex duality, this can be achieved by constructing a separating linear functional called a pseudo-moment certificate. We present constructive procedures to compute such certificates with rational coefficients for several famous forms (homogeneous polynomials) that are known to be positive but not SOS. Our method leverages polynomial symmetries to reduce the problem size and provides explicit integer-based formulas for generating these rational certificates. As a by-product, we can also generate extreme rays of the pseudomoment cone in exact arithmetic.

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arXiv2025

Polynomial argmin for recovery and approximation of multivariate discontinuous functions

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.

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