The role of numerical optimization has been continuously growing in the design of highfrequency structures, including microwave and antenna components. At the same time, accurate evaluation of electrical characteristics necessitates full-wave electromagnetic (EM) analysis, which is CPU intensive, especially for complex systems. As rigorous optimization routines involve repetitive EM simulations, the associated cost may be significant. In the design practice, the most widely used EM-driven procedures are by far local (e.g., gradient-based) ones. While typically incurring acceptable expenses that range from dozens to a few hundreds of objective function evaluations, they are prone to failure whenever a decent initial design is not available. Representative scenarios include simulation-based size reduction of compact devices or redesign of structures for operating/material parameters being distant from those at the available design. A standard mitigation approach is the involvement of global search methods, which entails significantly higher computational costs. This paper reviews the recent methodologies introduced to improve the reliability of local parameter tuning algorithms without degrading their computational efficiency. We discuss frequencybased regularization, adaptively adjusted design specification approach, as well as accelerated feature-based optimization. All of these techniques incorporate mechanisms that improve the performance of the search process under challenging scenarios, primarily poor initial conditions. The outline of the mentioned methods is accompanied by illustrative examples including passive microwave circuits and microstrip antennas. Benchmarking against conventional local search is provided as well. Furthermore, the paper discusses the advantages and disadvantages of the reviewed frameworks as well as speculates about future research directions.