A laplace duality for integration
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915275029348352 |
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| author | Lasserre, Jean B |
| author_facet | Lasserre, Jean B |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20842 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A laplace duality for integration Lasserre, Jean B Optimization and Control 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. |
| title | A laplace duality for integration |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2502.20842 |