Surrogate models are often used to reduce the cost of design optimization problems that involve computationally costly models, such as computational fluid dynamics simulations. However, the number of evaluations required by surrogate models usually scales poorly with the number of design variables, and there is a need for both better constraint formulations and multimodal function handling. To address this issue, we developed a surrogate-based gradient-free optimization algorithm that can handle cases where the function evaluations are expensive, the computational budget is limited, the functions are multimodal, and the optimization problem includes nonlinear equality or inequality constraints. The proposed algorithm-super efficient global optimization coupled with mixture of experts (SEGOMOE)can tackle complex constrained design optimization problems through the use of an enrichment strategy based on a mixture of experts coupled with adaptive surrogate models. The performance of this approach was evaluated for analytic constrained and unconstrained problems, as well as for a multimodal aerodynamic shape optimization problem with 17 design variables and an equality constraint. Our results showed that the method is efficient and that the optimum is much less dependent on the starting point than the conventional gradient-based optimization.

Bayesian optimization methods have been successfully applied to black box optimization problems that are expensive to evaluate. In this paper, we adapt the so-called super efficient global optimization algorithm to solve more accurately mixed constrained problems. The proposed approach handles constraints by means of upper trust bound, the latter encourages exploration of the feasible domain by combining the mean prediction and the associated uncertainty function given by the Gaussian processes. On top of that, a refinement procedure, based on a learning rate criterion, is introduced to enhance the exploitation and exploration trade-off. We show the good potential of the approach on a set of numerical experiments. Finally, we present an application to conceptual aircraft configuration upon which we show the superiority of the proposed approach compared to a set of the state-of-the-art black box optimization solvers.

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The multi-level, multidisciplinary and multi-fidelity optimization framework developed at Bombardier Aviation has shown great results to explore efficient and competitive aircraft configurations. This optimization framework has been developed within the Isight software, the latter offers a set of ready-to-use optimizers. Unfortunately, the computational effort required by the Isight optimizers can be prohibitive with respect to the requirements of an industrial context. In this paper, a constrained Bayesian optimization optimizer, namely the super efficient global optimization with mixture of experts, is used to reduce the optimization computational effort. The obtained results showed significant improvements compared to two of the popular Isight optimizers. The capabilities of the tested constrained Bayesian optimization solver are demonstrated on Bombardier research aircraft configuration study cases.