  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.
