Pseudo Expected Improvement Matrix Criteria for Parallel Expensive Multi-objective Optimization

Many engineering optimization problems involve multiple objectives which are sometimes computationally expensive. The multi-objective efficient global optimization (EGO) algorithm which uses a multi-objective expected improvement (EI) function as the infill criterion, is an efficient approach to solve these expensive multi-objective optimization problems. However, the state-of-the-art multi-objective EI criteria are very expensive to compute when the number of objectives is higher than two, thus are not practical to use in real-world problems. In the early work, the authors have proposed three cheap-to-calculate and yet efficient multi-objective EI matrix (EIM) criteria for the expensive multi-objective optimization. In this work, the three EIM criteria are extended for parallel computing to further accelerate the search process of the multi-objective EGO algorithm. The approach selects the first candidate at the maximum of an EIM criterion, and then multiplies the EIs in the EI matrix by the influence function of the first candidate to approximate the updated EIM function. The influence function is designed to simulate the effect that the first candidate will have on the landscape of each EI function. Then the second candidate can be selected at the maximum of the approximated EIM criterion. As the process goes on, a desired number of candidates can be generated in a single optimization iteration. The proposed parallel EIM (called pseudo EIM in this work) criteria have shown significant improvements over the single-point EIM criteria in terms of number of iterations on the selected test instances. The results indicate that the proposed pseudo EIM criteria can speed up the search process of the multi-objective EGO algorithm when parallel computing is available.