We describe a `discretize-then-relax' strategy to globally minimize integral functionals over functions $u$ in a Sobolev space satisfying prescribed 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-SOS 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 $L^p$ to the global minimizer of the original integral functional if this is unique. We also report computational experiments that show our numerical strategy works well even when technical conditions required by our theoretical analysis are not satisfied.

翻译：我们提出一种“先离散后松弛”策略，用于在满足给定狄利克雷边界条件的索伯列夫空间中，对函数$u$的积分泛函进行全局最小化。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（即使泛函非凸）。“离散化”步骤采用有界有限元格式，将积分最小化问题近似为紧致可行集上的一族收敛多项式优化问题，该族问题由递减的有限元网格尺寸$h$索引。“松弛”步骤则利用稀疏矩-SOS松弛，将每个多项式优化问题近似为一族凸半定规划，由递增的松弛阶数$\omega$索引。我们证明：当$\omega\to\infty$且$h\to 0$时，若原始积分泛函的全局最小解唯一，则这些半定规划的解所给出的近似最小化解在$L^p$意义下收敛至该全局最小解。此外，我们报告了数值实验，表明即便理论分析所需的技术条件未被满足，该数值策略仍能有效运行。