On the Use of Gradient-Enhanced Metamodels for Global Approximation and Global Optimization

International audienceIn the context of optimization, derivatives of the objective function or the constrained function with respect to the design parameters can be computed for increasing the performance of the optimisation algorithm. For dealing with mechanical optimisation, the first idea is to connect directly the mechanical solver with the optimizer. However, this approach is not conceivable in many cases because of the time consuming for obtaining the mechanical solution associated to a set of design parameters and the numerous iterations require by the optimizer for locating precisely the optimum. In particular the direct approach is unsuitable for dealing with costly industrial optimisation problems. Therefore, metamodels has been introduced for providing very inexpensive approximation of the objective and/or constrained function. Largely used since the 1950's many kinds of metamodels have been developped. They are built from a few actual responses providing by a solver associated to a set of sample points obtained by a sampling strategy. More recently many kinds of common metamodels has been extended for dealing with responses and gradients. Thus we propose to focus on these methods. Taking into account derivatives of the actual function allows us to built more accurate approximations with the same number of sample points [1, 2]. Thus indirect gradient-based metamodels, gradient-based Least-Square, Inverse Distance Weighting, Moving Least-Square, Weighting Least-Square, Radial Basis Function, Kriging and Support Vector Regression [3] are the most common gradient-based metamodels and they will be presented. The kernel-based metamodels (RBF, Kriging, SVR) will be more precisely detailed because they propose higher quality of approximation and better capacities for dealing with multimodal actual functions and higher number of design parameters in comparison with other techniques. Some comparisons of these approximation techniques will be proposed by considering global approximation criteria and global optimisation framework. In particular, the issue of the determination of the hyperparameters will be discussed. Factors of choice of well adapted approximation technique will be also proposed and discussed. In order to illustrate the performance of the gradient-based metamodels, many analytical functions will be considered with many number of design parameters. Finally, global optimisation of non-linear assembly problems [1, 2] will be considered by using a dedicated mechanical solver based on the MultiParametric Strategy and the LATIN method. The performance of the gradient-based metamodels optimisation algorithm will be also presented