Bayesian Quality-Diversity optimization for conditional search-space problems

The preliminary design and optimization of aerospace systems often requires to make architectural and technological choices and to perform computationally expensive numerical simulations. In early design phases, the decision makers may be interested in obtaining a set of valuable solutions presenting a diversity of characteristics. This paper proposes two approaches tackling these challenges by formulating and solving a particular optimization problem. The latter is called Conditional Search-Space Problem because architectural and technological choices are represented by design variables, called conditional, which are acting on the structure of the optimization problem (e.g., number of design variables and constraints). To produce a broad diversity of solutions, a Quality-Diversity (QD) algorithm adapted to deal with conditional variables and expensive functions is developed. The proposed methods consist in decomposing the optimization problem in several subproblems that have a non-conditional structure. A Bayesian optimization strategy coupled with a QD algorithm is used to search for potential solutions among the subproblems. The most performing potential solutions are then evaluated with the exact functions. The methodology is tested on two analytical and an aerospace system design problems.