Comparison Between Two Multi-Objective Optimization Algorithms: PAES and MGDA. Testing MGDA on Kriging Metamodels

Abstract

In multi-objective optimization, the knowledge of the Pareto set provides valuable information on the reachable optimal performance. A number of evolutionary strategies (PAES (Knowles and Corne in Evol. Comput. 8(2):149–172, 2000), NSGA-II (Deb et al. in IEEE Trans. Evol. Comput. 6(2):182–197, 2002), etc.), have been proposed in the literature and proved to be successful in identifying the Pareto set. However, these derivative-free algorithms are very demanding in computational time. Today, in many areas of computational sciences, codes are developed that include the calculation of the gradient, cautiously validated and calibrated. Thus, an alternate method applicable when the gradients are known is introduced presently. Using a clever combination of the gradients, a descent direction common to all criteria is identified. As a natural outcome, the Multiple Gradient Descent Algorithm (MGDA) is defined as a generalization of the steepest descent method and compared with the PAES by numerical experiments. Using the MGDA on a multi-objective optimization problem requires the evaluation of a large number of points with regard to criteria and their gradients. In the particular case of CFD problems, each point evaluation is very costly. Thus here we also propose to construct metamodels and to calculate approximate gradients by local finite differences.