Minmax robustness for multi-objective optimization problems

•Minmax robustness is extended to multi-objective optimization.•The concept of minmax robust efficiency is introduced.•Weighted sum scalarization and epsilon-constraints are adjusted to the new concept.•Problems with objective-wise uncertainty are investigated more closely.•The concepts are illustra...

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Published in:	European journal of operational research Vol. 239; no. 1; pp. 17 - 31
Main Authors:	Ehrgott, Matthias, Ide, Jonas, Schöbel, Anita
Format:	Journal Article
Language:	English
Published:	Amsterdam Elsevier B.V 16.11.2014 Elsevier Sequoia S.A
Subjects:	Disturbances Gain Ingredients Mathematical models Mathematical problems Mathematical programming Multi-objective optimization Operational research Optimization Optimization techniques Quadratic programming Quality Robustness Robustness and sensitivity analysis Scenarios Studies Uncertainty modelling Scenarios Multi-objective optimization Robustness and sensitivity analysis Uncertainty modelling
ISSN:	0377-2217, 1872-6860
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Abstract	•Minmax robustness is extended to multi-objective optimization.•The concept of minmax robust efficiency is introduced.•Weighted sum scalarization and epsilon-constraints are adjusted to the new concept.•Problems with objective-wise uncertainty are investigated more closely.•The concepts are illustrated with a few LP-examples. In real-world applications of optimization, optimal solutions are often of limited value, because disturbances of or changes to input data may diminish the quality of an optimal solution or even render it infeasible. One way to deal with uncertain input data is robust optimization, the aim of which is to find solutions which remain feasible and of good quality for all possible scenarios, i.e., realizations of the uncertain data. For single objective optimization, several definitions of robustness have been thoroughly analyzed and robust optimization methods have been developed. In this paper, we extend the concept of minmax robustness (Ben-Tal, Ghaoui, & Nemirovski, 2009) to multi-objective optimization and call this extension robust efficiency for uncertain multi-objective optimization problems. We use ingredients from robust (single objective) and (deterministic) multi-objective optimization to gain insight into the new area of robust multi-objective optimization. We analyze the new concept and discuss how robust solutions of multi-objective optimization problems may be computed. To this end, we use techniques from both robust (single objective) and (deterministic) multi-objective optimization. The new concepts are illustrated with some linear and quadratic programming instances.
AbstractList	In real-world applications of optimization, optimal solutions are often of limited value, because disturbances of or changes to input data may diminish the quality of an optimal solution or even render it infeasible. One way to deal with uncertain input data is robust optimization, the aim of which is to find solutions which remain feasible and of good quality for all possible scenarios, i.e., realizations of the uncertain data. For single objective optimization, several definitions of robustness have been thoroughly analyzed and robust optimization methods have been developed. In this paper, we extend the concept of minmax robustness (Ben-Tal, Ghaoui, & Nemirovski, 2009) to multi-objective optimization and call this extension robust efficiency for uncertain multi-objective optimization problems. We use ingredients from robust (single objective) and (deterministic) multi-objective optimization to gain insight into the new area of robust multi-objective optimization. We analyze the new concept and discuss how robust solutions of multi-objective optimization problems may be computed. To this end, we use techniques from both robust (single objective) and (deterministic) multi-objective optimization. The new concepts are illustrated with some linear and quadratic programming instances. •Minmax robustness is extended to multi-objective optimization.•The concept of minmax robust efficiency is introduced.•Weighted sum scalarization and epsilon-constraints are adjusted to the new concept.•Problems with objective-wise uncertainty are investigated more closely.•The concepts are illustrated with a few LP-examples. In real-world applications of optimization, optimal solutions are often of limited value, because disturbances of or changes to input data may diminish the quality of an optimal solution or even render it infeasible. One way to deal with uncertain input data is robust optimization, the aim of which is to find solutions which remain feasible and of good quality for all possible scenarios, i.e., realizations of the uncertain data. For single objective optimization, several definitions of robustness have been thoroughly analyzed and robust optimization methods have been developed. In this paper, we extend the concept of minmax robustness (Ben-Tal, Ghaoui, & Nemirovski, 2009) to multi-objective optimization and call this extension robust efficiency for uncertain multi-objective optimization problems. We use ingredients from robust (single objective) and (deterministic) multi-objective optimization to gain insight into the new area of robust multi-objective optimization. We analyze the new concept and discuss how robust solutions of multi-objective optimization problems may be computed. To this end, we use techniques from both robust (single objective) and (deterministic) multi-objective optimization. The new concepts are illustrated with some linear and quadratic programming instances.
Author	Ehrgott, Matthias Schöbel, Anita Ide, Jonas
Author_xml	– sequence: 1 givenname: Matthias surname: Ehrgott fullname: Ehrgott, Matthias email: m.ehrgott@lancaster.ac.uk organization: Lancaster University, Department of Management Science, Bailrigg, Lancaster LA1 4YX, United Kingdom – sequence: 2 givenname: Jonas surname: Ide fullname: Ide, Jonas email: j.ide@math.uni-goettingen.de organization: University of Göttingen, Institute for Numerical and Applied Mathematics, Lotzestr. 16-18, 37083 Göttingen, Germany – sequence: 3 givenname: Anita surname: Schöbel fullname: Schöbel, Anita email: schoebel@math.uni-goettingen.de organization: University of Göttingen, Institute for Numerical and Applied Mathematics, Lotzestr. 16-18, 37083 Göttingen, Germany
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Snippet	•Minmax robustness is extended to multi-objective optimization.•The concept of minmax robust efficiency is introduced.•Weighted sum scalarization and... In real-world applications of optimization, optimal solutions are often of limited value, because disturbances of or changes to input data may diminish the...
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SubjectTerms	Disturbances Gain Ingredients Mathematical models Mathematical problems Mathematical programming Multi-objective optimization Operational research Optimization Optimization techniques Quadratic programming Quality Robustness Robustness and sensitivity analysis Scenarios Studies Uncertainty modelling
Title	Minmax robustness for multi-objective optimization problems
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