Comptes Rendus
Article de recherche - Analyse numérique
Approximate D-optimal design and equilibrium measure
[Plans d’expérience D-optimaux et mesure d’équilibre]
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 739-756.

We introduce a minor variant of the approximate D-optimal design of experiments with a more general information matrix that takes into account the representation of the design space S. The main motivation (and result) is that if SRd is the unit ball, the unit box or the canonical simplex, then remarkably, for every dimension d and every degree n, one obtains an optimal solution in closed form, namely the equilibrium measure of S (in pluripotential theory). 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 finding an optimal design reduces to finding 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 semi-algebraic sets are also considered, and a previously developed two-step design algorithm is easily adapted to this new variant of D-optimal design problem.

Nous introduisons une variante mineure du plan d’expériences approché D-optimal avec une matrice d’information plus générale qui tient compte de la représentation de l’espace de conception S. La motivation principale (et le résultat principal) est que si SRd est la boule unitaire, la boîte unitaire ou le simplexe canonique, alors remarquablement, pour chaque dimension d et chaque degré n, on obtient une solution optimale explicite, à savoir la mesure d’équilibre de S (en théorie du pluripotentiel). De manière équivalente, pour chaque degré n, l’unique solution optimale est le vecteur des moments (jusqu’au degré 2n) de la mesure d’équilibre de S. Par conséquent, la recherche d’un modèle optimal se réduit à la recherche d’une cubature pour la mesure d’équilibre, avec des atomes dans S, des poids positifs, et des valeurs exactes jusqu’au degré 2n. En outre, toute séquence résultante de mesures atomiques D-optimales converge vers la mesure d’équilibre de S, pour la topologie étoile faible, pour n croissant. Le lien avec les ensembles de points de Fekete sont également discutés. Des ensembles semi-algébriques compacts basiques plus généraux sont également considérés, et un algorithme de conception en deux étapes développé précédemment est facilement adapté à cette nouvelle variante du plan de conception D-optimal.

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DOI : 10.5802/crmath.766
Classification : 62K05, 32U15, 90C22

Didier Henrion 1, 2, 3 ; Jean-Bernard Lasserre 1, 3

1 LAAS-CNRS, BP 54200, 7 avenue du Colonel Roche, 31031 Toulouse cedex 4, France
2 Czech Technical University, Praha, Czech Republic
3 Toulouse School of Economics (TSE), Toulouse, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Approximate {D-optimal} design and equilibrium measure},
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Didier Henrion; Jean-Bernard Lasserre. Approximate D-optimal design and equilibrium measure. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 739-756. doi : 10.5802/crmath.766. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.766/

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