Engineering optimization procedures employ highly accurate numerical models that typically have an excessive computational cost, e.g., finite elements (FE). The space mapping (SM) technique speeds up the minimization procedure by exploiting simplified (less accurate) models. We will use the SM terminology of fine and coarse to refer to the accurate and inaccurate models, respectively. SM implementation in the field of electromagnetic actuators design, in the context of constrained optimization, is a rather unexplored topic. A linear voice coil actuator is chosen as a benchmark test example. The key element in SM is the so-called SM function, which efficiently corrects the imprecise results that can be obtained with just coarse information. SM is used to solve a shape optimization problem. The design problem is stated as a minimization in which the computational cost lies completely in the constraint evaluation; thus, SM is applied only to the constraints. A mathematical description of the approach is presented in this paper and two implementations are compared. The solution of the SM optimization is validated (locally) by means of a standard minimization routine. The numerical results obtained show high efficiency of the SM-based optimization algorithm, reflected by a significantly low number of fine model simulations and an overall low computational effort.