GLODS-SI-Mix: A Deterministic Direct-Search Method for Mixed-Variable Black-Box Optimization

Preprint version of a manuscript currently under peer review. This version has not been peer reviewed and is not the final published version. We propose GLODS-SI-Mix, a deterministic direct-search method for bound-constrained mixed-variable black-box optimization with continuous, ordered discrete, and categorical variables. The method extends the GLODS-SI two-space principle to mixed-variable domains: polling, cache lookup, distance comparisons, and list-management operations are performed in a canonical search space, while objective-function evaluations are carried out only after reconstruction in the original variables. The canonical state stores continuous variables as normalized coordinates, ordered discrete variables as indices, and categorical variables as permutation-invariant identity labels. A single step-size parameter controls all poll moves. Categorical variables are handled through Deterministic Neighborhood Rotation, which explores categorical alternatives deterministically without imposing a fixed metric, artificial ordering, or preliminary learning phase. The convergence statement is conservative: on stabilized continuous fibres, accumulation points satisfy first-order nonnegativity along feasible refining directions, while finite-valued components are treated through local rank-one neighbourhood consistency statements. Computational experiments on mixed-variable benchmark problems and a scaling study provide evidence of deterministic reproducibility and scale-invariant behaviour.