Exploring polynomial models in the Search Step of Direct Multisearch

Abstract: Direct Multisearch (DMS) is a class of direct-search algorithms designed for multiobjective derivative-free optimization. Its framework consists of an optional search step and a poll step, the latter ensuring the corresponding theoretical convergence properties. Recently, a search strategy based on the minimization of quadratic polynomial models, constructed from previously evaluated points, was proposed to improve the numerical efficiency of the method. While the construction of these surrogate models has been investigated, considerably less attention has been devoted to how they should be jointly minimized, with min-max scalarization typically being adopted. This work investigates how different model-minimization strategies influence the performance of DMS. To this end, alternative strategies for exploring the quadratic polynomial models within the search step of DMS are proposed and numerically assessed, including one based on the recently proposed Improved Front Steepest Descent algorithm.