Pointwise expected hypervolume improvement for expensive multi-objective optimization

The expected hypervolume improvement (EHVI) is a popular infill criterion used in the multi-objective efficient global optimization algorithm for solving expensive multi-objective optimization problems. However, its exact calculation is complex and time-consuming when the number of objectives is larger than three, which prohibits its usage in problems with more than three objectives. To tackle this problem, we propose a new simple and fast criterion called pointwise expected hypervolume improvement (pEHVI) in this work. In the proposed criterion, we first compute the EHVI values of the studying point by considering one non-dominated front point at a time, and then take the minimum of these EHVI values as the final measurement. The proposed pEHVI is derived in closed-form expression and has linear time complexity with respect to the number of objectives. In addition, we theoretically prove the monotonicity and convergence properties of the proposed pEHVI criterion in this work. Compared with the traditional expected hypervolume improvement, the new infill criterion is significantly faster to compute, especially when the number of objectives is larger than three. Numerical experiments show that the proposed criterion is also able to achieve competitive optimization performance compared with five state-of-the-art multi-objective efficient global optimization algorithms. This work provides a fast and efficient hypervolume-based expected improvement infill criterion for expensive multi-objective optimization.