Search Techniques for Multi-Objective Optimization of Mixed-Variable Systems Having Stochastic Responses


A research approach is presented that extends the separate solution methods of stochastic and multi-objective optimization problems to one that would solve problems having both characteristics. Such problems are typically encountered when one desires to optimize systems with multiple, often competing, objectives that do not have a closed form representation and must be estimated, e.g., via simulation. First, the class of mesh adaptive direct search (MADS) algorithms for nonlinearly constrained optimization is extended to mixed variable problems, and convergence to appropriately defined stationary points is proved. The resulting algorithm, MVMADS, is then extended to stochastic problems (MVMADS-RS), via a ranking and selection procedure. Finally, a two-stage method is developed that combines the generalized pattern search/ranking and selection (MGPS-RS) algorithms for singleobjective, mixed variable, stochastic problems with a multi-objective approach that makes use of interactive techniques for the specification of aspiration and reservation levels, scalarization functions, and multi-objective ranking and selection. This combination is devised specifically so as to keep the desirable convergence properties of MGPS-RS and MVMADS-RS, while extending to the multi-objective case. A convergence analysis for the general class of algorithms establishes almost sure convergence of an iteration subsequence to stationary points appropriately defined in the mixedvariable domain. Seven specific instances of the new algorithm are implemented and tested on 11 multi-objective test problems from the literature and an engineering design problem.