arXiv subjects

Jean B Lasserre

Publications and source records attributed to Jean B Lasserre.

8 recordsLinked to original sources

arXiv2026

Rank conditions for exactness of semidefinite relaxations in polynomial optimization

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 $ϕ$ n and a semi-algebraic set K $\subset$ 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 $ϕ$ 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.

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arXiv2026

Connecting max-entropy with computational geometry, LP and SDP

We consider the well-known max-(relative) entropy problem $Θ$(y) = infQ$\ll$P DKL(Q P ) with Kullback-Leibler divergence on a domain $Ω$ $\subset$ R d , and with ''moment'' constraints h dQ = y, y $\in$ R m . We show that when m $\le$ d, $Θ$ is the Cram{é}r transform of a function v that solves a simply related computational geometry problem. Also, and remarkably, to the canonical LP: min x$\ge$0 {c T x\,: A x = y}, with A $\in$ R mxd , one may associate a max-entropy problem with a suitably chosen reference measure P on R d + and linear mapping h(x) = Ax, such that its associated perspective function $ε$ $Θ$(y/$ε$) is the optimal value of the log-barrier formulation (with parameter $ε$) of the dual LP (and so it converges to the LP optimal value as $ε$ $\rightarrow$ 0). An analogous result also holds for the canonical SDP: min X 0 { C, X\,: A(X) = y }.

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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

Gaussian-like fixed point and variational properties of integral discriminants

We consider partition functions Z(g) = exp (-g(x))dx where g is a nonnegative polynomial action (a degree-2n form) vanishing only at the origin. Such integrals, known as integral discriminants, appear in statistical mechanics, quantum field theory, and the theory of exponential families. We show that the associated Boltzmann measure d$μ$ = exp(-g(x))dx satisfies a fixed-point property identity relating in a simple manner its degree-2n moments to the coefficients of g. This generalizes familiar identities for the exponential distribution (degree-1) on the positive orthant and the Gaussian measure (degree-2). We further show that g is characterized by three variational principles, including a maximum-entropy principle under scaled moments constraints, extending the Gaussian extremality principle to arbitrary even-degree homogeneous actions. Exploiting these identities in a truncatedmoment numerical scheme (known as the Moment-SOS hierarchy), strengthens the standard semidefinite relaxations, and results in a much faster convergence, thus allowing more efficient approximations of the partition function Z(g) as well as moments of $μ$.

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arXiv2025

An extension of the mean value theorem

Let ($Ω$, $μ$) be a measure space with $Ω$ $\subset$ R d and $μ$ a finite measure on $Ω$. We provide an extension of the Mean Value Theorem (MVT) in the form It is valid for non compact sets $Ω$ and f is only required to be integrable with respect to $μ$. It also contains as a special case the MVT in the form f d$μ$ = $μ$($Ω$)f (x 0 ) for some x 0 $\in$ $Ω$, valid for compact connected set $Ω$ and continuous f . It is a direct consequence of Richter's theorem which in turn is a non trivial (overlooked) generalization of Tchakaloff's theorem, and even published earlier.

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arXiv2025

A laplace duality for integration

We consider the integral v(y) = Ky f (x)dx on a domain Ky = {x $\in$ R d\,: g(x) $\le$ y}, where g is nonnegative and Ky is compact for all y $\in$ [0, +$\infty$). Under some assumptions, we show that for every y $\in$ (0, $\infty$) there exists a distinguished scalar $λ$y $\in$ (0, +$\infty$) such that which is the counterpart analogue for integration of Lagrangian duality for optimization. A crucial ingredient is the Laplace transform, the analogue for integration of Legendre-Fenchel transform in optimization. In particular, if both f and g are positively homogeneous then $λ$y is a simple explicitly rational function of y. In addition if g is quadratic form then computing v(y) reduces to computing the integral of f with respect to a specific Gaussian measure for which exact and approximate numerical methods (e.g. cubatures) are available.

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arXiv2025

Shannon-and von neumann-entropy regularizations of linear and semidefinite programs

We consider the LP in standard form min {c T x\,: Ax = b; x $\ge$ 0} and inspired by $ε$-regularization in Optimal Transport, we introduce its $ε$-regularization ''min {c T x + $ε$ f (x)\,: Ax = b; x $\ge$ 0}'' via the (convex) Boltzmann-Shannon entropy f (x)\,:= i x i ln x i . We also provide a similar regularization for the semidefinite program ''min {Tr(C $\bullet$ X)\,: A(X) = b; X 0}'' but with now the so-called Von Neumann entropy, as in Quantum Optimal Transport. Importantly, both are not barriers of the LP and SDP cones respectively. We show that this problem admits an equivalent unconstrained convex problem max $λ$$\in$R m G$ε$($λ$) for an explicit concave differentiable function G$ε$ in dual variables $λ$ $\in$ R m . As $ε$ goes to zero, its optimal value converges to the optimal value of the initial LP. While it resembles the log-barrier formulation of interior point algorithm for the initial LP, it has a distinguishing advantage. Namely for fixed $λ$, G$ε$($λ$) is obtained as a minimization over the whole space x $\in$ R d (and not over x $\ge$ 0) to still obtain a nonnegative solution x($λ$) $\ge$ 0, whence an explicit form of G$ε$ very useful for its unconstrained maximization over R m .

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arXiv2022

Stokes, Gibbs and volume computation of semi-algebraic sets

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).

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