A new predictor corrector variant for unconstrained bi-objective optimization problems

Abstract

Many real-life applications can be formulated as a multi-objective optimization problem. Since the solutions set, the Pareto set, of such problems typically forms locally a manifold, specialized predictor-corrector methods that allow to follow the solution set from a given optimal point have proven to be very effective for such problems. In this work, we investigate two different possible choices for the predictor direction for bi-objective problems: One that was recently proposed in literature and another one which is a variant of the ‘classical’ choice adapted to multi-objective optimization but has the advantage that it does not require the consideration of the weight space coming from the Karush-Kuhn-Tucker formulation in an augmented system. From our observations we derive three new continuation variants and compare them on some benchmark models. The numerical results indicate that the novel choice of the predictor that uses the Hessians of the objectives leads to some savings in the computational effort compared to the classical continuation method, and that its Hessian free realization via Quasi-Newton methods leads to further significant improvements in the overall cost in terms of function evaluations.