The Pareto Tracer for the treatment of degenerated multi-objective optimization problems

Multi-objective continuation algorithms are very powerful tools for the numerical treatment of continuous multi-objective optimization problems (MOPs). All of these methods, however, are based on certain regularity assumptions that imply that the solution set of the given MOP forms, at least locally, objects of below a certain dimension. While this is indeed the case for most problems, there exist examples where the Pareto set/front is lower-dimensional, which are called degenerated cases. This article presents and discusses a new predictor step designed for use within multi-objective continuation methods that automatically detects (numerical) degeneration and performs movements in essential directions. Furthermore, this predictor is integrated into the multi-objective continuation method 'Pareto Tracer'. The effectiveness of this new continuation method is demonstrated on selected benchmark problems, illustrating its capability to handle both degenerated and non-degenerated MOPs efficiently.