Projektdaten

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Zur Stützeigenschaft in der multikriteriellen Optimierung

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Although weighted sum scalarization is a popular method to find solutions of multiobjective optimization problems, it is well-known that in general not all solutions can be found this way, but only the supported ones. A standard assumption for guaranteeing the supportedness of al solutions is the convexity of the multiobjective problem. The first main aim of this project is to identify a significantly larger class than convex multiobjective problems, for which all solutions are supported and can thus be computed by weighted sum scalarization. We call such multiobjective optimization problems supported. Since the essential consequence of the above convexity assumption is the convexity of the so-called upper image set, also hidden convex problems possess the desired property, by which we mean problems which possess a convex upper image set despite being nonconvex. Moreover, a multiobjective problem may even be supported although the upper image set is nonconvex. We aim at a better understanding of the class of supported multiobjective optimization problems which, in the above sense, is a category between convex and general nonconvex multiobjective optimization. In this context we also study relaxation techniques, like copositive or semidefinite reformulations, for multiobjective optimization. We aim on understanding which relaxations may be promising for multiobjective optimization and which are tight in the supported solutions only and can also be obtained by first scalarizing with a weighted sum and then applying single-objective relaxation techniques. The second main aim of this project bases on the observation that supportedness of solutions of multiobjective problems is not an intrinsic property but may be enforced by a certain image space transformation technique for some or even for all solutions. After a thorough study of this technique, we aim at designing algorithmically feasible constructions which move multiobjective problems with nonsupported sol