Nonlinear optimization of mixed continuous and discrete variables for black-box simulators

In recent years, there has been a considerable number of industrial applications that involve mixed variables and time-consuming simulators, e.g., at Safran Tech and IFPEN: optimal designs of aircraft engine turbine, of mooring lines of offshore wind turbines, of electric engine statorsn and rotors,. . . In these nonlinear optimization problems, derivatives of the objective function (and, possibly of the constraint functions) are not available and cannot be directly approximated. Another difficulty is that these problems involve heterogeneous nature variables: a varying number of components (integer variables), different materials (categorical variables, usually nonordered), the presence or not of some components (binary variables), and continuous variables describing dimensions/characteristics of the structure pieces. This thesis aims to develop and adapt Derivative-Free Optimization (DFO) methods for different types of applications, including the optimal design of aircraft engines. In the first part, we focus on the development and adaptation of a DFO method to problems with continuous and mixed discrete variables exhibiting a cyclic-symmetry property. For that purpose, we introduce the necklace distance and tailor accordingly the trust-region constraints of the optimization problems. Before running our adapted method on a simplified simulation provided by Safran, we build a set of benchmark functions by transforming them into a set of cyclic-symmetry test functions. We run our method on these benchmark functions and on a Safran’s simulated instance with a large number of repetitions to study the robustness of the method compared to other state-of-the-art methods. We also give a local convergence proof of our adapted method. In the second part, we focus on the design of experiments in mixed continuous and discrete variables space by extending the kernel-embedding distribution from continuous space to mixed discrete variables case. This part of the thesis is motivated by the need to improve the initialization phase of the optimization algorithm with a better exploration of the space of mixed variables, guided by the available prior information (types of variables, symmetry, correlations, ...). We illustrate the potential of the proposed approach in the more classical framework of meta-model function approximation for continuous and discrete mixed variables, and also for time series. Finally, we give ideas to improve the proposed optimization method for a better exploration of the space of design variables to avoid being trapped in local minima.