On Continuation Methods for Non-Linear Bi-Objective Optimization: Certified Interval-Based Approach

The global optimization of constrained Non-Linear Bi-Objective Optimization problems (NLBOO) aims at covering their Pareto-optimal front which is in general a manifold in R. Continuation methods can help in this context as they can follow a continuous component of this front once an initial point on it is provided. They constitute somehow a generalization of the classical scalarizing framework which transforms the bi-objective problem into a parametric monoobjective problem. Recent works have shown that they can play a key role in global algorithms dedicated to bi-objective problems, e.g. population based algorithms, where they allow discovering large portions of locally Pareto optimal vectors, which turns out to strongly support diversification. In this paper, we provide a survey on continuation techniques in global optimization methods for NLBOO, which allow discovering large portions of locally Pareto-optimal solutions. We also propose a rigorous active set management strategy on top of a previously proposed certified continuationmethod based on interval analysis, and illustrate it on a challenging bi-objective problem.