Escaping Unknown Discontinuous Regions in Blackbox Optimization

Abstract

The design of key nonlinear systems often requires the use of expensive blackbox simulations presenting inherent discontinuities whose positions in the variable space cannot be analytically predicted. Without further precautions, the solution of related optimization problems leads to design configurations which may be close to discontinuities of the blackbox output functions. These discontinuities may betray unsafe regions of the design space, such as nonlinear resonance regions. To account for possible changes of operating conditions, an acceptable solution must be away from unsafe regions of the space of variables. The objective of this work is to solve a constrained blackbox optimization problem with the additional constraint that the solution should be outside unknown zones of discontinuities or strong variations of the objective function or the constraints. The proposed approach is an extension of the mesh adaptive direct search (\sf Mads) algorithm and aims at building a series of inner approximations of these zones. The algorithm, called \sf DiscoMADS, relies on two main mechanisms: revealing discontinuities and progressively escaping the surrounding zones. A convergence analysis supports the algorithm and preserves the optimality conditions of \sf Mads. Numerical tests are conducted on analytical problems and on three engineering problems illustrating the following possible applications of the algorithm: the design of a simplified truss, the synthesis of a chemical component, and the design of a turbomachine blade. The \sf DiscoMADS algorithm successfully solves these problems by providing a feasible solution away from discontinuous regions.