SO-MI: A surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems

Abstract

This paper introduces a surrogate model based algorithm for computationally expensive mixed-integer black-box global optimization problems with both binary and non-binary integer variables that may have computationally expensive constraints. The goal is to find accurate solutions with relatively few function evaluations. A radial basis function surrogate model (response surface) is used to select candidates for integer and continuous decision variable points at which the computationally expensive objective and constraint functions are to be evaluated. In every iteration multiple new points are selected based on different methods, and the function evaluations are done in parallel. The algorithm converges to the global optimum almost surely. The performance of this new algorithm, SO-MI, is compared to a branch and bound algorithm for nonlinear problems, a genetic algorithm, and the NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search) algorithm for mixed-integer problems on 16 test problems from the literature (constrained, unconstrained, unimodal and multimodal problems), as well as on two application problems arising from structural optimization, and three application problems from optimal reliability design. The numerical experiments show that SO-MI reaches significantly better results than the other algorithms when the number of function evaluations is very restricted (200–300 evaluations).