Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations

Abstract

Multi-objective optimization algorithms aim at finding Pareto-optimal solutions. Recovering Pareto 
fronts or Pareto sets from a limited number of function evaluations are challenging problems. A popular 
approach in the case of expensive-to-evaluate functions is to appeal to metamodels. Kriging has been shown 
efficient as a base for sequential multi-objective optimization, notably through infill sampling criteria balancing 
exploitation and exploration such as the Expected Hypervolume Improvement. Here we consider Kriging 
metamodels not only for selecting new points, but as a tool for estimating the whole Pareto front and quantifying 
how much uncertainty remains on it at any stage of Kriging-based multi-objective optimization algorithms. Our 
approach relies on the Gaussian process interpretation of Kriging, and bases upon conditional simulations. Using 
concepts from random set theory, we propose to adapt the Vorob'ev expectation and deviation to capture the 
variability of the set of non-dominated points. Numerical experiments illustrate the potential of the proposed 
workflow, and it is shown on examples how Gaussian process simulations and the estimated Vorob'ev deviation 
can be used to monitor the ability of Kriging-based multi-objective optimization algorithms to accurately learn the Pareto front.