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

(Preprint version of a manuscript currently under peer review. This version has not been peer reviewed and is not the final published version.) Many multiobjective optimization problems in computational operations research, simulation-based decision making, and engineering design involve black-box functions and decision variables of different types. Continuous variables have a natural local geometry, ordered discrete variables have an ordinal structure, and categorical variables are purely nominal. Applying direct-search ideas to such problems therefore requires a search geometry that is meaningful across variable types and insensitive to arbitrary choices of scaling, spacing, or labeling. We propose DMS-SI-Mix, a deterministic direct multisearch framework for mixed-variable multiobjective derivative-free optimization. The method keeps the standard DMS mechanism of maintaining a nondominated list and selecting poll centers from that list, but performs polling in a canonical search space; objective functions are evaluated only after reconstruction in the original decision space. A single step-size parameter controls the continuous, ordered discrete, and categorical poll moves, with a common fraction-of-domain interpretation across variable types. This separation makes the canonical search trajectory invariant under representation changes that carry no algorithmic information: affine rescaling of continuous variables, numerical spacing of ordered discrete variables, and arbitrary relabeling of categorical variables. The convergence analysis is deliberately local and conditional. Under the stated assumptions and along refining subsequences, accumulation points satisfy a mixed-block Pareto stationarity condition: Pareto–Clarke stationarity along the continuous polling directions, together with weak local Pareto optimality on the ordered discrete and categorical blocks, each expressed in its native geometry. Computational results illustrate the practical effect of the canonical formulation. On continuous benchmark problems with heterogeneous scaling, and on an engineering piezo-shunt design problem, the scale-invariant formulation yields statistically significant improvements over Direct Multisearch under heterogeneous scaling, while behaving comparably under uniform scaling. On controlled mixed-variable benchmark pairs, the method preserves meaningful Pareto-front structure after reformulation, with problem-dependent degradation when the finite blocks restrict the attainable front. The resulting framework provides a simple and reproducible way of extending direct multisearch to heterogeneous decision spaces without imposing a global metric on them.