Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations

• Published in: European journal of operational research 2015-06, Vol.243 (2), p.386-394
• Main Authors: Binois, M., Ginsbourger, D., Roustant, O.
• Format: Article
• Language: English
• Subjects: Attainment function; Expected Hypervolume Improvement; Kriging; Machine Learning; Mathematics; Methodology; Multi-objective optimization; Operations research; Optimization algorithms; Optimization and Control; Pareto optimum; Set theory; Simulation; Statistics; Stochastic models; Studies; Vorob’ev expectation
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2014.07.032

•We study the ability of Kriging-based algorithms to learn the Pareto front.•Random sets theory is applied based on conditional Gaussian process simulations.•Estimation and visualization of the uncertainty on Pareto fronts is provided.•Uncertainty measures allow defining stopping criteria in a sequential framework.•The interest of the proposed methodology is illustrated on two example problems. 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.