Pareto Tracer : A Predictor Corrector Method for Multi - objective Optimization Problems

In many real-world applications, the problem arises that several objectives have to be optimized concurrently leading to a multi-objective optimization problem. Since these goals are typically contradictory, it comes as no surprise that the solution set— the so-called Pareto set—of such problems does (in general) not consist of one single solution. Moreover, under some mild conditions, this set is typically a smooth manifold (like a surface or solid). For the decision maker it is hence desired to obtain a suitable finite size representation of the optimal set, or to explore (locally) the possibilities around a given solution in situations where the entire Pareto set is too large. In this thesis, we devise a novel predictor corrector method to address multiobjective optimization problems by continuation. The algorithm, Pareto Tracer, is capable to trace the manifold of (local) Pareto solutions of a problem with in principle any number of objectives. Our proposal can cope with box and equality constraints for which a Newton-like method was designed to deal with restrictions. Furthermore, two Hessian-free realizations of the Pareto Tracer were developed based respectively on quasi-Newton and gradient descent approaches. We discuss the algorithm first theoretically and demonstrate further on its strength on several examples.