Motivated by optimization with differential equations, we consider optimization problems with Hilbert spaces as decision spaces. As a consequence of their infinite dimensionality, the numerical solution necessitates finite dimensional approximations and discretizations. We develop an approximation framework and demonstrate criticality measure-based error estimates. We consider criticality measures inspired by those used within optimization methods, such as semismooth Newton and (conditional) gradient methods. Furthermore, we show that our error estimates are order-optimal. Our findings augment existing distance-based error estimates, but do not rely on strong convexity or second-order sufficient optimality conditions. Moreover, our error estimates can be used for code verification and validation. We illustrate our theoretical convergence rates on linear, semilinear, and bilinear PDE-constrained optimization.

翻译：受微分方程优化的启发，我们考虑以希尔伯特空间为决策空间的优化问题。由于其无穷维特性，数值求解需要有限维近似和离散化。我们建立了一个近似框架，并展示了基于临界性度量的误差估计。我们考虑了受优化方法（如半光滑牛顿法和（条件）梯度法）启发而建立的临界性度量。此外，我们证明我们的误差估计是最优阶的。我们的发现补充了现有的基于距离的误差估计，但不依赖于强凸性或二阶充分最优性条件。此外，我们的误差估计可用于代码验证与确认。我们在线性、半线性和双线性偏微分方程约束优化问题上展示了理论收敛速度。