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 $\omega\to\infty$ and $h\to 0$, 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.

翻译：我们描述了一种“先离散再松弛”的策略，用于在Sobolev空间中满足Dirichlet边界条件的函数$u$上全局极小化积分泛函。该策略适用于积分泛函关于$u$及其导数呈多项式依赖的情形，即使其非凸。“离散”步骤采用有界有限元格式，通过一系列在紧可行集上的多项式优化问题的收敛层级来近似积分极小化问题，这些层级由有限元网格的递减尺寸$h$索引。“松弛”步骤则利用稀疏矩-平方和松弛，通过一系列凸半定规划的层级来近似每个多项式优化问题，这些层级由递增的松弛阶数$\omega$索引。我们证明，当$\omega\to\infty$且$h\to 0$时，这些半定规划的解提供的近似极小值点会在适当意义下（包括某些$L^p$范数）收敛到原积分泛函的全局极小值点（若其唯一）。我们还报告了计算实验，显示即使理论分析所需的技术条件未被满足，我们的数值策略依然表现良好。