A variational approach to define robustness for parametric multiobjective optimization problems |
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Authors: | Katrin Witting Sina Ober-Blöbaum Michael Dellnitz |
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Institution: | 1. Chair of Applied Mathematics, University of Paderborn, Warburger Str. 100, 33098, Paderborn, Germany 2. Computational Dynamics and Optimal Control, University of Paderborn, Warburger Str. 100, 33098, Paderborn, Germany
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Abstract: | In contrast to classical optimization problems, in multiobjective optimization several objective functions are considered at the same time. For these problems, the solution is not a single optimum but a set of optimal compromises, the so-called Pareto set. In this work, we consider multiobjective optimization problems that additionally depend on an external parameter ${\lambda \in \mathbb{R}}$ , so-called parametric multiobjective optimization problems. The solution of such a problem is given by the λ-dependent Pareto set. In this work we give a new definition that allows to characterize λ-robust Pareto points, meaning points which hardly vary under the variation of the parameter λ. To describe this task mathematically, we make use of the classical calculus of variations. A system of differential algebraic equations will turn out to describe λ-robust solutions. For the numerical solution of these equations concepts of the discrete calculus of variations are used. The new robustness concept is illustrated by numerical examples. |
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