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：我们描述了一种“先离散后松弛”的策略，用于在满足给定狄利克雷边界条件的Sobolev空间中，对函数$u$的积分泛函进行全局极小化。该策略适用于积分泛函关于$u$及其导数为多项式形式（即使非凸）的情形。“离散”步骤采用有界有限元格式，将积分极小化问题近似为一族定义在紧致可行集上的多项式优化问题，该族问题具有收敛性，并由有限元网格尺寸$h$的递减指数标记。“松弛”步骤则利用稀疏矩-SOS松弛方法，将每个多项式优化问题近似为一族凸半定规划问题，该族问题由递增的松弛阶数$\omega$指数标记。我们证明，当$\omega\to\infty$且$h\to 0$时，若原始积分泛函的全局极小点唯一，则这些半定规划问题的解所提供的近似极小点将在$L^p$意义下收敛至该全局极小点。此外，我们报告了计算实验结果，表明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有效。