A Trust-Region Algorithm for Bi-Objective Stochastic Optimization

Abstract

We develop a new method for approximating the Pareto front of a bi-objective stochastic optimization problem in which the expected objective functions are estimated by taking sample averaged outputs from expensive simulations. At each iteration of the proposed algorithm, a trust region is identified and quadratic approximate functions for the expected objective functions are built using the sample average values. To determine non-dominated solutions in the trust region, a single-objective optimization problem is constructed based on the approximate objective functions. After updating the set of non-dominated solutions, a new trust region around the most isolated point is determined to explore areas that have not been visited. When the computational budget is limited, a large sample size at each iteration leads to more accurate approximation of the expected objective functions, but the algorithm is not able to run for enough iterations to generate a set of solutions that are close to the Pareto front. The proposed variable sampling scheme adaptively updates the sample size with consideration for this trade-offetween approximation and optimization errors. The numerical results show that our proposed method is feasible, and the performance can be significantly improved with an appropriate sampling scheme.