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Numerical simulation methods at the fields level, such as the finite-element method, are highly accurate but computationally expensive. In the context of mathematical optimization, this implies that the cost function, which needs to be evaluated a large number of times, is very expensive to compute. To overcome this shortcoming, this paper proposes to employ parametric reduced-order models for computing the cost function and, if required, its gradients. They introduce low systematic error, require little memory, and allow for hundreds to thousands of model evaluations per second. The great utility of the suggested approach in both deterministic and stochastic optimization methods is demonstrated by a numerical example, featuring 11 geometric parameters.