A simplex-like search method for bi-objective optimization

Abstract

We describe a new algorithm for bi-objective optimization, similar to the Nelder Mead simplex algorithm, widely used for single objective optimization. For differentiable bi-objective functions on a continuous search space, internal Pareto optima occur where the two gradient vectors point in opposite directions. So such optima may be located by minimizing the cosine of the angle between these vectors. This requires a complex rather than than a simplex, so we term the technique the “cosine seeking complex”. An extra benefit of this approach is that a successful search identifies the direction of the efficient curve of Pareto points, expediting further searches. Results are presented for some standard test functions. The method presented is quite complicated and space considerations here preclude complete details. We hope to publish a fuller description in another place