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.

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