RESEARCH PAPER Pareto surface construction for multi-objective optimization under uncertainty Chen Liang 1 & Sankaran Mahadevan 1 Received: 20 October 2015 /Revised: 19 July 2016 /Accepted: 22 September 2016 /Published online: 2 December 2016 # Springer-Verlag Berlin Heidelberg 2016 Abstract This paper presents a novel approach for multi- objective optimization under both aleatory and epistemic sources of uncertainty. Given paired samples of the inputs and outputs from the system analysis model, a Bayesian network (BN) is built to represent the joint probability distribution of the inputs and outputs. In each design iteration, the optimizer provides the values of the design variables to the BN, and copula-based sampling is used to rapidly generate samples of the output variables conditioned on the input values. Samples from the conditional distributions are used to evaluate the objec- tives and constraints, which are fed back to the optimizer for further iteration. The proposed approach is formulated in the context of reliability-based design optimization (RBDO). The joint probability of multiple objectives and constraints is includ- ed in the formulation. The Bayesian network along with condi- tional sampling is exploited to select training points that enable effective construction of the Pareto front. A vehicle side impact problem is employed to demonstrate the proposed methodology. Keywords Multi-objective optimization . Uncertainty quantification . Bayesian network . Gaussian copula Nomenclature x design variables p non-design variables lb x , ub x lower and upper bounds of x f i i th deterministic objective g i i th deterministic constraint μ f i mean of the i th objective P jt joint probability constraint CDF cumulative distribution function PDF probability density function DV design variable UV uncertain variable obj objective constr constraint F i (x i ) marginal CDF of the i th random variable x i C copula function Φ − 1 inverse CDF of a standard normal random variable R covariance matrix of Gaussian copula I identity matrix L likelihood function L ab Abdomen load Rib u Upper rib deflection Rib m Middle rib deflection Rib l Lower rib deflection VC u Upper viscous criteria VC m Middle viscous criteria VC l Lower viscous criteria F Pb Pubic force Con i i th constraint of the vehicle side impact example Crit criterion of the i th constraint 1 Introduction In multi-objective optimization (MOO) with competing objec- tives, the multiple solutions are often characterized through a Pareto surface, which is a series of designs describing the tradeoff among different objectives. The decision maker will select the appropriate design alternative based on his/her pref- erences on the objectives (Marler and Arora 2004). Four types * Sankaran Mahadevan sankaran.mahadevan@vanderbilt.edu 1 Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN, USA Struct Multidisc Optim (2017) 55:1865–1882 DOI 10.1007/s00158-016-1619-7