A Two-Space Scale-Invariant Direct Multisearch Method for Mixed-Variable Multiobjective Optimization

Abstract

Many multiobjective black-box problems combine continuous, ordered-discrete, and categorical variables. Direct search on such domains requires a geometry that respects each variable type and is insensitive to arbitrary scaling, spacing, and labeling. We propose DMS-SI-Mix, a deterministic two-space Direct Multisearch method. The decision-side algorithmic state and poll geometry are defined in a canonical search space, whereas every trial is reconstructed into the original decision space before the black-box model is evaluated. Black-box evaluators therefore receive the original physical or problem-defined variables. A single resolution parameter controls canonical continuous moves, ordered-discrete rank moves, and categorical moves generated by deterministic neighborhood rotation. This canonical-search/original-evaluation separation yields invariance of the canonical trajectory under affine continuous rescaling, order-preserving discrete relabeling, and consistent bijective renaming of categorical identities. Under the stated assumptions, the persistent center of any refining subsequence satisfies a conditional mixed-block stationarity result: Pareto-Clarke stationarity along the continuous polling directions and weak local Pareto optimality on the finite blocks. On 108 continuous benchmarks, the recorded two-space configuration attains significantly larger Hypervolume than original-coordinate DMS under every tested heterogeneous scaling, while no systematic difference is detected under uniform scaling. On six mixed reformulations with up to ten categorical variables and 126 identities per variable, CC-DNR preserves the nominal poll size of a fixed geometry, whereas full one-coordinate polling generates nominal polls of 175-820 trial slots. On MORAP-NM, with an exact 578-point Pareto front, CC-DNR recovers 575 exact Pareto decisions and matches the final objective archive of full polling at lower poll cost.