Browsing by Author "Fantuzzi, Giovanni"

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    GLOBAL MINIMIZATION OF POLYNOMIAL INTEGRAL FUNCTIONALS
    (2024) Fantuzzi, Giovanni; Fuentes, Federico
    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 omega . We prove that, as omega ->infinity and h -> 0 , solutions of such semidefinite programs provide approximate minimizers that converge in a suitable sense (including in certain L-p 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.