Pseudo expected improvement criterion for parallel EGO algorithm

The efficient global optimization (EGO) algorithm is famous for its high efficiency in solving computationally expensive optimization problems. However, the expected improvement (EI) criterion used for picking up candidate points in the EGO process produces only one design point per optimization cycle, which is time-wasting when parallel computing can be used. In this work, a new criterion called pseudo expected improvement (PEI) is proposed for developing parallel EGO algorithms. In each cycle, the first updating point is selected by the initial EI function. After that, the PEI function is built to approximate the real updated EI function by multiplying the initial EI function by an influence function of the updating point. The influence function is designed to simulate the impact that the updating point will have on the EI function, and is only corresponding to the position of the updating point (not the function value of the updating point). Therefore, the next updating point can be identified by maximizing the PEI function without evaluating the first updating point. As the sequential process goes on, a desired number of updating points can be selected by the PEI criterion within one optimization cycle. The efficiency of the proposed PEI criterion is validated by six benchmarks with dimension from 2 to 6. The results show that the proposed PEI algorithm performs significantly better than the standard EGO algorithm, and gains significant improvements over five of the six test problems compared against a state-of-the-art parallel EGO algorithm. Furthermore, additional experiments show that it affects the convergence of the proposed algorithm significantly when the global maximum of the PEI function is not found. It is recommended to use as much evaluations as one can afford to find the global maximum of the PEI function.