When Scaling Is Not Neutral: Parameterization Effects in Simulation-Driven Nelder–Mead Optimization of Microwave Filters

Design-variable scaling is often treated as numerical preprocessing in simulation-driven optimization, although the coordinates supplied to the optimizer form part of the numerical interface to the computational model. This study quantifies the resulting implementation-level sensitivity of Nelder–Mead using two full-wave electromagnetic (EM) microwave-filter models as controlled engineering benchmarks: a three-variable low-pass filter (LPF) with comparable variable magnitudes and a five-variable band-pass filter (BPF) containing a much smaller coupling gap. For each benchmark, the physical seeds, EM model, frequency samples, sampled mask-based objective, aggregation norm, optimizer options, and evaluation budget were fixed; only the optimizer-space representation was changed. Raw physical, relative-normalized, partially scaled gap, and log-relative coordinates were compared using exact and tolerance-based success, residual violation severity, EM evaluation count, seed-level outcomes, and final-design distances. For the LPF, raw and normalized coordinates each achieved 5/15 exact successes, while log-relative coordinates achieved 4/15 with milder residual violations. For the BPF, the corresponding counts were 13/15, 13/15, 12/15, and 9/15, with the log-relative formulation producing the most severe failures. The results show that parameterization is a non-neutral component of simulation-driven optimization and should be documented and validated as part of reproducible computational-modeling workflows.