Electromagnetic (EM) analysis is the primary tool utilized in the design of high-frequency structures. In vast majority of cases, simpler models (e.g., equivalent networks or analytical ones) are either not available or lack accuracy: they can only be used to yield initial designs that need to be further tuned. Consequently, EM-driven adjustment of geometry and/or material parameters of microwave and antenna components is a necessary design stage. This, however, is a computationally expensive process, not only because of a considerable computational cost of high-fidelity EM analysis but also due to a typically large number of parameters that need to be adjusted. In particular, conventional numerical optimization routines (both local and global) may be prohibitively expensive. In this paper, a reduced-cost trust-region-based gradient search algorithm is proposed for the optimization of high-frequency components. Our methodology is based on a smart management of the system Jacobian enhancement which combines: (i) omission of (finite-differentiation-based) sensitivity updates for variables that exhibit small (relative) relocation in the directions of the corresponding coordinate system axes and (ii) selective utilization of a rank-one Broyden updating formula. Parameter selection for Broyden-based updating depends on the alignment between the direction of the latest design relocation and respective search space basis vectors. The proposed technique is demonstrated using a miniaturized coupler and an ultra-wideband antenna. In both cases, significant reduction of the number of EM simulations involved in the optimization process is achieved as compared to the benchmark algorithm (computational speedup of 60% on average). At the same time, degradation of the design quality is minor.