Tracing the Pareto frontier in bi-objective optimization problems by ODE techniques

Abstract

In this paper we present a deterministic method for tracing the Pareto frontier in non-linear bi-objective optimization problems with equality and inequality constraints. We reformulate the bi-objective optimization problem as a parametric single-objective optimization problem with an additional Normalized Normal Equality Constraint (NNEC) similar to the existing Normal Boundary Intersection (NBI) and the Normalized Normal Constraint method (NNC). By computing the so called Defining Initial Value Problem (DIVP) for segments of the Pareto front and solving a continuation problem with a standard integrator for ordinary differential equations (ODE) we can trace the Pareto front. We call the resulting approach ODE NNEC method and demonstrate numerically that it can yield the entire Pareto frontier to high accuracy. Moreover, due to event detection capabilities available for common ODE integrators, changes in the active constraints can be automatically detected. The features of the current algorithm are illustrated for two case studies whose Matlab® code is available as Electronic Supplementary Material to this article.