Global optimization of expensive black box functions using potential Lipschitz constants and response surfaces

Abstract

This article develops a novel global optimization algorithm using potential Lipschitz constants and response surfaces (PLRS) for computationally expensive black box functions. With the usage of the metamodeling techniques, PLRS proposes a new approximate function $${\hat{F}}$$F^ to describe the lower bounds of the real function $$f$$f in a compact way, i.e., making the approximate function $${\hat{F}}$$F^ closer to $$f$$f. By adjusting a parameter $${\hat{K}}$$K^ (an estimate of the Lipschitz constant $$K$$K), $${\hat{F}}$$F^ could approximate $$f$$f in a fine way to favor local exploitation in some interesting regions; $${\hat{F}}$$F^ can also approximate $$f$$f in a coarse way to favor global exploration over the entire domain. When doing optimization, PLRS cycles through a set of identified potential estimates of the Lipschitz constant to construct the approximate function from fine to coarse. Consequently, the optimization operates at both local and global levels. Comparative studies with several global optimization algorithms on 53 test functions and an engineering application indicate that the proposed algorithm is promising for expensive black box functions.