Constrained Optimization in Simulation: Efficient Global Optimization and Karush-Kuhn-Tucker Conditions

An important goal of simulation is optimization of the corresponding real system. We focus on simulation with multiple responses, selecting one response as the variable to be minimized while the remaining responses satisfy prespecified thresholds: so-called constrained optimization. We treat the simulation model as a black box. We assume that the simulation is computationally expensive; therefore, we use an inexpensive metamodel (emulator, surrogate) of the simulation model. A popular metamodel type is a Kriging or Gaussian process (GP) model (GP is also used in supervised learning). For optimization with a single response, this GP is used in efficient global optimization (EGO) (and also in Bayesian optimization, which is related to active learning).We develop an innovative EGO variant for constrained deterministic optimization where the optimal solution lies on one or more binding (input or output) constraints; therefore, we use the Karush-Kuhn-Tucker (KKT) conditions. We combine these conditions with the expected improvement (EI) criterion, which is popular in EGO. To evaluate the performance of our variant, we apply it to three popular examples (namely, one mathematical and two engineering design problems); these examples show promising numerical results when compared with other recent methods.