arXiv subjects

Jean B. Lasserre

Publications and source records attributed to Jean B. Lasserre.

At least 19 recordsLinked to original sources

arXiv2026

Finite convergence of the Moment-SOS hierarchy under hidden convexity

We consider polynomial optimization problems with compact feasible set ${\bfΩ}\subset\mathbb{R}^d$ defined by SOS-concave polynomials $g_j$ of arbitrary degree, and whose objective function $f$ is not necessarily convex on $\mathbb{R}^d$. We show that, if $f$ is Hessian-$Q$-module convex over ${\bfΩ}$ in the sense that its Hessian admits a specific quadratic-module representation, then the standard Moment-SOS hierarchy converges in finitely many steps without prior knowledge of this hidden (local) convexity. Strong convexity of $f$ on ${\bfΩ}$ is a sufficient condition for the required Hessian representation. In addition, we give an explicit relaxation order at which exactness occurs. This demonstrates that a general-purpose hierarchy can adapt to favorable hidden properties of a specific instance without being informed of them, yielding certified global minimizers.

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arXiv2022

Certifying Global Optimality of AC-OPF Solutions via sparse polynomial optimization

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.

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arXiv2022

Non-negative forms, volumes of sublevel sets, complete monotonicity and moment matrices

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.

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arXiv2021

CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization

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.

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arXiv2015

Moments and Legendre-Fourier Series for Measures Supported on Curves

Some important problems (e.g., in optimal transport and optimal control) have a relaxed (or weak) formulation in a space of appropriate measures whichis much easier to solve. However, an optimal solution $μ$ of the latter solves the former if and only if the measure $μ$ is supported on a "trajectory" $\{(t,x(t))\colon t\in [0,T]\}$ for some measurable function $x(t)$. We provide necessary and sufficient conditions on moments $(γ\_{ij})$ of a measure $dμ(x,t)$ on $[0,1]^2$ to ensure that $μ$ is supported on a trajectory $\{(t,x(t))\colon t\in [0,1]\}$. Those conditions are stated in terms of Legendre-Fourier coefficients ${\mathbf f}\_j=({\mathbf f}\_j(i))$ associated with some functions $f\_j\colon [0,1]\to {\mathbb R}$, $j=1,\ldots$, where each ${\mathbf f}\_j$ is obtained from the moments $γ\_{ji}$, $i=0,1,\ldots$, of $μ$.

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arXiv2014

Volume of slices and sections of the simplex in closed form

Given a vector a $\in$ Rn, we provide an alternative and direct proof for the formula of the volume of sections delta $\cap$ {x : a T x \textless{}= t} and slices $\cap$ {x : a T x = t}, t $\in$ R, of the simplex delta. For slices the formula has already been derived but as a by-product of the construction of univariate B-Splines. One goal of the paper is to also show how simple and powerful can be the Laplace transform technique to derive closed form expression for some multivariate integrals. It also complements some previous results obtained for the hypercube [0, 1] n .

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arXiv2014

Tractable approximations of sets defined with quantifiers

Given a compact basic semi-algebraic set $K\subset R^n\times R^m$, a simple set $B$ (box or ellipsoid), and some semi-algebraic function $f$, we consider sets defined with quantifiers, of the form $R_f:=\{x\in B: \mbox{$f(x,y)\leq 0$ for all $y$ such that $(x,y)\in K$}\}$ and $D_f:=\{x\in B: \mbox{$f(x,y)\geq 0$ for some $y$ such that $(x,y)\in K$}\}$. The former set $R_f$ is particularly useful to qualify "robust" decisions $x$ versus noise parameter $y$ (e.g. in robust optimization on some set $\mathbfΩ\subset B$) whereas the latter set $D_f$ is useful (e.g. in optimization) when one does not want to work with its lifted representation $\{(x,y)\in K: f(x,y)\geq 0\}$. Assuming that $K_x:=\{y:(x,y)\in K\}\neq\emptyset$ for every $x\in B$, we provide a systematic procedure to obtain a sequence of explicit inner (resp. outer) approximations that converge to $R_f$ (resp. $D_f$) in a strong sense. Another (and remarkable) feature is that each approximation is the sublevel set of a single polynomial whose vector of coefficients is an optimal solution of a semidefinite program. Several extensions are also proposed, and in particular, approximations for sets of the form $R_F:=\{x\in B:\mbox{$(x,y)\in F$ for all $y$ such that $(x,y)\in K$}\}$, where $F$ is some other basic-semi algebraic set, and also sets defined with two quantifiers.

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arXiv2012

Exploiting symmetries in SDP-relaxations for polynomial optimization

In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute in the geometric quotient.

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arXiv2011

A new look at nonnegativity on closed sets and polynomial optimization

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $dν=fdμ$ with support equal to K, are all positive semidefinite (if $K$ is compact $μ$ is an arbitrary finite Borel measure with support equal to K. In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with {\it no} lifting, of the cone of nonnegative polynomials of degree at most $d$. Wen used in polynomial optimization on certain simple closed sets $\K$ (like e.g., the whole space $\R^n$, the positive orthant, a box, a simplex, or the vertices of the hypercube), it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable. This convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations.

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arXiv2010

Lp-norms, Log-barriers and Cramer transform in Optimization

We show that the Laplace approximation of a supremum by Lp-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem P and its dual appear naturally when using this simple approximation technique for the value function g of P or its Legendre-Fenchel conjugate. In addition, minimizing the LBF of the dual is just evaluating the Cramer transform of the Laplace approximation of g. Finally, this technique permits to sometimes define an explicit dual problem in cases when the Legendre-Fenchel conjugate of g cannot be derived explicitly from its definition.

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arXiv2010

A "joint+marginal" algorithm for polynomial optimization

We present a new algorithm for solving a polynomial program P based on the recent "joint + marginal" approach of the first author for, parametric optimization. The idea is to first consider the variable x1 as a parameter and solve the associated (n-1)-variable (x2,...,xn) problem P(x1) where the parameter x1 is fixed and takes values in some interval Y1 with some probability uniformly distributed on Y1. Then one considers the hierarchy of what we call "joint+marginal" semidefinite relaxations, whose duals provide a sequence of univariate polynomial approximations that converges to the optimal value function J(x1) of problem P(x1), as k increases. Then with k fixed a priori, one computes a minimizer of the univariate polynomial pk(x1) on the interval Y1, which reduces to solving a single semidefinite program. One iterates the procedure with now an (n-2)-variable problem P(x2) with parameter x2 in some new interval Y2, etc. The quality of the approximation depends on how large k can be chosen (in general for significant size problems, k=1 is the only choice). Preliminary numerical results are provided

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arXiv2010

Certificates of convexity for basic semi-algebraic sets

We provide two certificates of convexity for arbitrary basic semi-algebraic sets of $\R^n$. The first one is based on a necessary and sufficient condition whereas the second one is based on a sufficient (but simpler) condition only. Both certificates are obtained from any feasible solution of a related semidefinite program and so can be obtained numerically (however, up to machine precision).

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arXiv2009

Semidefinite Programming for Min-Max Problems and Games

We introduce two min-max problems: the first problem is to minimize the supremum of finitely many rational functions over a compact basic semi-algebraic set whereas the second problem is a 2-player zero-sum polynomial game in randomized strategies and with compact basic semi-algebraic pure strategy sets. It is proved that their optimal solution can be approximated by solving a hierarchy of semidefinite relaxations, in the spirit of the moment approach developed in Lasserre. This provides a unified approach and a class of algorithms to approximate all Nash equilibria and min-max strategies of many static and dynamic games. Each semidefinite relaxation can be solved in time which is polynomial in its input size and practice from global optimization suggests that very often few relaxations are needed for a good approximation (and sometimes even finite convergence).

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arXiv2009

On representations of the feasible set in convex optimization

We consider the convex optimization problem $\min \{f(x) : g_j(x)\leq 0, j=1,...,m\}$ where $f$ is convex, the feasible set K is convex and Slater's condition holds, but the functions $g_j$ are not necessarily convex. We show that for any representation of K that satisfies a mild nondegeneracy assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely every KKT point is a minimizer. That is, the KKT optimality conditions are necessary and sufficient as in convex programming where one assumes that the $g_j$ are convex. So in convex optimization, and as far as one is concerned with KKT points, what really matters is the geometry of K and not so much its representation.

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arXiv2009

A "joint+marginal" approach to parametric polynomial optimization

Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_y$) on $R^n$ whose description depends on the parameter $y\inY$. We assume that one can compute all moments of some probability measure $ϕ$ on $Y$, absolutely continuous with respect to the Lebesgue measure (e.g. $Y$ is a box or a simplex and $ϕ$ is uniformly distributed). We then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions $x^*(y)$ of $P_y$. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions, like e.g. their $ϕ$-mean. In addition, using this knowledge on moments, the measurable function $y\mapsto x^*_k(y)$ of the $k$-th coordinate of optimal solutions, can be estimated, e.g. by maximum entropy methods. Also, for a boolean variable $x_k$, one may approximate as closely as desired its persistency $ϕ(\{y:x^*_k(y)=1\})$, i.e. the probability that in an optimal solution $x^*(y)$, the coordinate $x^*_k(y)$ takes the value 1. At last but not least, from an optimal solution of the dual semidefinite relaxations, one provides a sequence of polynomial (resp. piecewise polynomial) lower approximations with $L_1(ϕ)$ (resp. almost uniform) convergence to the optimal value function.

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arXiv2008

Convexity in semi-algebraic geometry and polynomial optimization

We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide a certificate of convexity if a sufficient (and almost necessary) condition is satified. This condition can be checked numerically and also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of recent results from the author and Helton and Nie. Finally, we show that when restricting to a certain class of convex polynomials, the celebrated Jensen's inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures.

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arXiv2008

Measures with zeros in the inverse of their moment matrix

We investigate and discuss when the inverse of a multivariate truncated moment matrix of a measure $μ$ has zeros in some prescribed entries. We describe precisely which pattern of these zeroes corresponds to independence, namely, the measure having a product structure. A more refined finding is that the key factor forcing a zero entry in this inverse matrix is a certain conditional triangularity property of the orthogonal polynomials associated with $μ$.

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arXiv2008

Representation of nonnegative convex polynomials

We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic subset K of Rn. Namely, they belong to a specific subset of the quadratic module generated by the concave polynomials that define K.

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