Oliver Schütze, Cinvestav-IPN in Mexico City, Mexico.
Abstract: In many applications, the problem arises that several conflicting objectives have to be optimized concurrently.
One important characteristic of such multi-objective optimization problems (MOPs) is that their solution sets are typically not given by a singleton as for “classical” scalar optimization, but form a manifold of a certain dimension.
In this talk, we will present subdivision and continuation methods for the numerical treatment of MOPs. Subdivision algorithms start with a compact subset of the domain, represented by a collection of boxes. In each iteration step of the algorithm, the boxes in the current collection are subdivided, and it is decided which of these have to be kept for further iterations. The underlying algorithm converges toward the relative global attractor of used dynamical system(s) in the Hausdorff sense. Algorithms of this type are of global nature and have demonstrated their strength on complicated moderate dimensional problems. On the other hand, the global approach prevents their applicability to high-dimensional problems.
In the second part of this talk, we will present the multi-objective continuation methods Pareto Tracer (PT) and Pareto Explorer (PE) that take advantage of the fact that the solution set of a MOP forms — at least locally — a manifold. While PT and PE are of local nature, they are are applicable to high-dimensional problems regarding both the dimension of the decision variable space as well as the number of objectives. These continuation methods can handle general constraints of the model, and are capable of the efficient treatment of degenerated problems that represent a challenge to many other solvers.