Article Dans Une Revue IEEE Control Systems Letters Année : 2025

A laplace duality for integration

Résumé

We consider the integral v(y) = Ky f (x)dx on a domain Ky = {x ∈ R d : g(x) ≤ y}, where g is nonnegative and Ky is compact for all y ∈ [0, +∞). Under some assumptions, we show that for every y ∈ (0, ∞) there exists a distinguished scalar λy ∈ (0, +∞) 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 λy 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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Dates et versions

hal-04970360 , version 1 (27-02-2025)
hal-04970360 , version 2 (05-03-2025)
hal-04970360 , version 3 (03-05-2025)

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Jean B Lasserre. A laplace duality for integration. IEEE Control Systems Letters, 2025, 9, pp.168 - 173. ⟨10.1109/LCSYS.2025.3567851⟩. ⟨hal-04970360v3⟩
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