Biobjective optimization using direct search techniques

A new method is proposed for solving biobjective optimization problems by adaptively filling the gaps in a Pareto front. The proposed method deals with bound constrained biobjective optimization of possibly nonsmooth functions for problems where the nature of the objective functions is not known apriori or the function evaluation is very expensive (e.g., computer simulation based function evaluations). The proposed method employs a hybrid optimization approach using two direct search techniques (dividing rectangles and mesh adaptive direct search) and an adaptive weighting scheme for converting a biobjective optimization problem into a single objective optimization problem. The significant contributions of this paper include the novel idea of using the optimization algorithm DIRECT as a sampling strategy and the introduction of a precise mathematical measure, star discrepancy, as a measure of assessing the distribution of the obtained Pareto optimal solutions. Results show that the proposed Pareto approximation algorithm provides an arbitrarily close approximation to the true Pareto front and obtains a well dispersed global Pareto optimal set for diverse types of Pareto fronts (convex, nonconvex, and discontinuous).