"Enclosures and local upper bounds in multiobjective optimization"Eichfelder, GabrieleIn single-objective global optimization, numerical algorithms generally aim to stop with an interval with a predefined length which contains the optimal value. Transferring this idea to optimization problems with vector-valued objective functions, i.e. to multiobjective optimization, leads to the concept of enclosures. This concept has proven to be very helpful, not only as stopping criterion but also for discarding tests in mixed-integer programming or for adaptive refinement techniques for nonconvex functions. We present the concept of an enclosure which is often constructed using so-called local upper and lower bounds. The theory for these local upper and lower bounds is meanwhile well understood and makes heavily use of the componentwise structure of the natural ordering in an m-dimensional space. The talk also answers the question whether and how these concepts can be generalized to more general partial orders defined by polyhedral ordering cones. We define local upper bounds with respect to a closed pointed solid convex cone and study their properties. We show that for special polyhedral ordering cones the concept of local upper bounds can be as practical as it is for the nonnegative orthant. This talk is based on joint work with Firdevs Ulus, Bilkent University. |
« back