Global minimization of polynomial integral functionals
Fantuzzi G, Fuentes F (2024)
Publication Language: English
Publication Status: Submitted
Publication Type: Journal article, Online publication
Future Publication Type: Journal article
Publication year: 2024
Journal
Book Volume: 46
Journal Issue: 4
DOI: 10.1137/23M1592584
Open Access Link: https://arxiv.org/abs/2305.18801
Abstract
We describe a “discretize-then-relax” strategy to globally minimize integral functionals over functions u in a Sobolev space subject to Dirichlet boundary conditions. The strategy applies whenever the integral functional depends polynomially on u and its derivatives, even if it is nonconvex. The “discretize” step uses a bounded finite element scheme to approximate the integral minimization problem with a convergent hierarchy of polynomial optimization problems over a compact feasible set, indexed by the decreasing size h of the finite element mesh. The “relax” step employs sparse moment-sum-of-squares relaxations to approximate each polynomial optimization problem with a hierarchy of convex semidefinite programs, indexed by an increasing relaxation order ω. We prove that, as ω tends to infinity and h tends to zero, solutions of such semidefinite programs provide approximate minimizers that converge in a suitable sense (including in certain Lp norms) to the global minimizer of the original integral functional if it is unique. We also report computational experiments showing that our numerical strategy works well even when technical conditions required by our theoretical analysis are not satisfied.
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APA:
Fantuzzi, G., & Fuentes, F. (2024). Global minimization of polynomial integral functionals. SIAM Journal on Scientific Computing, 46(4). https://doi.org/10.1137/23M1592584
MLA:
Fantuzzi, Giovanni, and Federico Fuentes. "Global minimization of polynomial integral functionals." SIAM Journal on Scientific Computing 46.4 (2024).
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