José F A Madeira
Abstract Engineering design optimization frequently involves decision variables spanning disparate physical scales, such as spatial coordinates in millimetres coexisting with electrical resistances in kilohms, or dimensionless material fractions alongside dimensional thicknesses. Under such heterogeneity, derivative-free optimizers that rely on absolute distances for algorithmic decisions tend to degrade, as step sizes and proximity tests become biased toward large-range variables. This paper presents GLODS-SI, a scale-invariant reformulation of the GLODS (global and local optimization using direct search) derivative-free global optimization framework. Rather than a pre-processing normalization, GLODS-SI adopts a two-space algorithmic formulation: all geometric operations—trial-point generation, polling, distance computations, and merging decisions—are performed in normalized coordinates $[0,1]^n$, while objective evaluations are carried out in the original variables. This decoupling ensures that step-size and merging parameters acquire an intrinsic, dimensionless meaning independent of variable units and ranges, eliminating the need for problem-specific parameter tuning. The reformulation preserves all convergence guarantees of the original GLODS framework, and introduces a reproducible benchmarking methodology for evaluating scale sensitivity, including seven deterministic scaling strategies with contrasts up to $10^8$. Numerical experiments on 63 benchmark problems show that GLODS-SI maintains consistently high success rates (84–91%) across all scaling regimes, whereas the original formulation degrades to 70–84% under heterogeneous scales. A comparison with NOMAD, a widely used derivative-free solver, further confirms the competitiveness of GLODS-SI under the full evaluation budget. The scaling toolkit and benchmark suite are publicly available; the complete GLODS-SI implementation will be released upon acceptance.