15h30 - 15h55
On generalized-convex constrained multi-objective optimization
This talk is devoted to the study of general multi-objective optimization
problems involving convex and nonconvex constraints. The vector-valued
objective function of the considered multi-objective optimization problem
is assumed to be componentwise generalized-convex (e.g., semi-strictly
quasi-convex or quasi-convex). We show that the set of efficient solutions (in an
arbitrarily normed space) of a multi-objective optimization problem involving
a nonempty closed (not necessarily convex) feasible set, can be computed completely
using two corresponding multi-objective optimization problems with a
new feasible set that is a convex upper set of the original feasible set. Our
approach relies on the fact that the original feasible set can be described using
level sets of a certain scalar function (a kind of penalization function). At the
end of the talk we apply our approach to problems where the constraints are
given by a system of inequalities with a finite number of constraint functions.
15h55 - 16h20
Necessary optimality conditions for some nonconvex facility location problems
The problem of locating a new facility with simultaneous consideration of existing attraction and repulsion points is a problem with great practical relevance e.g. in the fields of economy, city planning or industrial design. Unfortunately, the consideration of negative weights makes this problem in general a nonconvex one, so that none of the established algorithms for location problems are able to solve it.
We will therefore present a new approach to derive necessary optimality conditions for such problems using the nonconvex subdifferentials by Ioffe and Kruger/Mordukhovich. While there are many strong theoretical results on these subdifferentials, it is rarely possible to explicitly calculate them or use them for applications.
After giving a brief review on definition, properties and calculus of the mentioned subdifferentials we will show, that for certain distance functions it is possible to precisely calculate the corresponding subdifferentials. By taking advantage of the special structure of the problems we can then derive necessary optimality conditions for some instances of scalar semi-obnoxious facility location problems. Furthermore, we will present some new scalarization results and use them to establish necessary optimality conditions for multicriteria semi-obnoxious facility location problems. At the end of the talk, we will give an outlook on open questions and possible future developments.
16h20 - 16h45
A linear vector programming approach to global quasi-concave problems
In the case of a bounded polyhedral feasible region it is well known that a quasi-concave function attains its global minimum in a vertex. This property enables us to apply a version of Benson’s algorithm for solving the minimization problem. Ehrgott and Shao introduced a variant of Benson’s algorithm to solve special linear multiplicative problems under polyhedral constraints. With respect to certain assumptions, a generalized algorithm is formulated to solve a wider range of problems. We also improve the suggested dual algorithm. Ultimately, we apply recent polyhedral projection results to weaken an important assumption.
16h45 - 17h10
alphaBB Method for Global Multiobjective Optimization
A well-known deterministic method in smooth scalar-valued global optimization is the alpha Branch and Bound (alphaBB) method which uses convex underestimators of the objective function and a partition of the search domain. We use the technique of convex underestimators for deriving also a method for solving multiobjective optimization problems globally. Thereby, a first step, which is already known in the literature, is to use the ideal point of the underestimated multiobjective problem for pruning. We improve this discarding test by using ideas from outer approximation techniques from convex multiobjective optimization combined with the concept of local upper bounds.