ParEGO Extensions for Multi-objective Optimization of Expensive Evaluation Function

International audienceThis paper deals with multi-objective optimization in the case of expensive objective functions. Such problem arises in a lot of engineering applications, including for instance automotive system design where the main purpose is to find a set of optimal solutions in a limited global processing time. Widely used algorithms for multi-objective optimization are ParEGO (Pareto Efficient Global Optimization) and NSGA-II (Nondominated Sorting Genetic Algorithm). NSGA-II is an evolutionary algorithm that recursively sorts the set of parameter values according to a measurement of Rank and Crowing distance and updates the parameter values using a genetic algorithm. In practice, such algorithm may require a high population size and several population updates, especially in the case of high-dimensional problems and more than two objectives functions. ParEGO algorithm is an extension of the mono-objective global optimization method called EGO. Its use for multi-objective optimization consists in combining linearly all the objectives with several random weights and to maximize the expected improvement (EI) criterion, which is based on a surrogate model obtained by Kriging. In high dimensions, ParEGO algorithm tends to favor parameter values suitable for the reduction of the surrogate model error, rather than finding the non-dominated solutions. The contribution of this article is to propose an extension of the ParEGO algorithm for finding the Pareto Front by introducing a double Kriging strategy allowing to calculate a modified EI criterion that jointly accounts for the objective function approximation error and the probability to find Pareto Front solutions. The main feature of the resulting algorithm is to enhance the convergence speed and thus to reduce the total number of function evaluations; which is of high interest in high-dimensional or expensive evaluations problems. The performances of this new algorithm are compared with the methods ParEGO and NSGA-II on a standard benchmark problem (ZDT functions). Finally, the use of the algorithm in an automotive engineering application is presented : the parameter setting optimization of the controller and the state observer of the auto-steer system in the autonomous car desig