A kriging based method for the solution of mixed-integer nonlinear programs containing black-box functions

In this paper a new methodology is developed for the solution of mixed–integer nonlinear programs under uncertainty whose problem formulation is complicated by both noisy variables and black–box functions representing a lack of model equations. A branch–and–bound framework is employed to handle the integer complexity whereby the solution to the relaxed nonlinear program subproblem at each node is obtained using both global and local information. Global information is obtained using kriging models which are used to identify promising neighborhoods for local search. Response surface methodology (RSM) is then employed whereby local models are sequentially optimized to refine the problem's lower and upper bounds. This work extends the capabilities of a previously developed kriging–response surface method enabling a wider class of problems to be addressed containing integer decisions and black box models. The proposed algorithm is applied to several small process synthesis examples and its effectiveness is evaluated in terms of the number of function calls required, number of times the global optimum is attained, and computational time.