Singularities in Multi-Objective Optimization and their Crossing during Continuation

Abstract

Continuation methods help trace Pareto sets in multi-objective optimization but are inherently local: a single run traces a single connected branch, requiring multiple restarts to recover disconnected components of Pareto fronts. We show that, for unconstrained bi-objective problems under weighted-sum scalarization, these disconnects can be artifacts of singularities in the scalarization parameter, where the weight $λ$ diverges as the objective gradients become collinear. Recasting Pareto optimality as a nonlinear system, we apply pseudo-arclength continuation to follow the Pareto-critical set, and show that suitable singular reparameterizations allow crossing these singularities in systematically, recovering disconnected branches in a single run. A coordinate-wise projective compactification further provides a unified framework for parameter and decision-space variables. We demonstrate the approach on the ZDT3 benchmark and modifications.