Expected improvement for expensive optimization: a review

The expected improvement (EI) algorithm is a very popular method for expensive optimization problems. In the past twenty years, the EI criterion has been extended to deal with a wide range of expensive optimization problems. This paper gives a comprehensive review of the EI extensions designed for parallel optimization, multiobjective optimization, constrained optimization, noisy optimization, multi-fidelity optimization and high-dimensional optimization. The main challenges of extending the EI approach to solve these complex optimization problems are pointed out, and the ideas proposed in literature to tackle these challenges are highlighted. For each reviewed algorithm, the surrogate modeling method, the computation of the infill criterion and the internal optimization of the infill criterion are carefully studied and compared. In addition, the monotonicity properties of the multiobjective EI criteria and constrained EI criteria are analyzed in detail. Through this review, we give an organized summary about the EI developments in the past twenty years and show a clear picture about how the EI approach has advanced. In the end of this paper, several interesting problems and future research topics about the EI developments are given.