Multifidelity Quasi-Newton Method for Design Optimization

Abstract

Multifidelity approaches are frequently used in design when high-fidelity models are too expensive to use directly and lower-fidelity models of reasonable accuracy exist. In optimization, corrected low-fidelity data are typically used in a sequence of independent, approximate optimizations bounded by trust regions. A new, unified, multifidelity quasi-Newton approach is presented that preserves an approximate inverse Hessian between iterations, determines search directions from high-fidelity data, and uses low-fidelity models for line searches. The proposed algorithm produces better search directions, maintains larger step sizes, and requires significantly fewer low-fidelity function evaluations than Trust Region Model Management. The multifidelity quasi-Newton method also provides an expected optimal point that is forward looking and is useful in building superior low-fidelity corrections. The new approach is compared with Trust Region Model Management and the BFGS quasi-Newton method on several analytic test problems using polynomial and kriging corrections. For comparison, a technique is demonstrated to initialize high-fidelity optimization when transition away from approximate models is deemed fruitful. In summary, the unified multifidelity quasi-Newton approach required fewer or equal high-fidelity function evaluations than Trust Region Model Management in about two-thirds of the test cases, and similarly reduced cost in more than half of cases compared with BFGS.