On robustness for set-valued optimization problems
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On robustness for set-valued optimization problems Kuntal Som1
· V. Vetrivel1
Received: 24 August 2019 / Accepted: 5 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In the recent past, finding robust solutions for optimization problems contaminated with uncertainties has been topical and has been investigated in the literature for scalar and multiobjective/vector-valued optimization problems. In this paper, we introduce various types of robustness concept for set-valued optimization, such as min–max set robustness, optimistic set robustness, highly set robustness, flimsily set robustness, multi-scenario set robustness. We study some existence results for corresponding concepts of solution and establish some relationship among them. Keywords Set-valued optimization · Robustness · Uncertainty
1 Introduction Set-valued optimization has become a vibrant area of research with many applications such as in risk management [9,10], multi-criteria decision making, social choice theory [25], statistics [12] and others. There exist many different concepts of solution for a set-valued optimization problem based on different approaches, such as the vector approach, the set approach, the lattice approach, the embedding approach, etc. One can refer to [11,16–20,23,24] for studies related to set-valued optimization. On the other hand, robust optimization has been a topic of much interest in the optimization community after the seminal work of Ben-Tal et al. [2,3]. Actually, most of the real-life optimization problems suffer from uncertainties, especially when they are very sensitive to small data perturbation and therefore need solutions that take uncertainties into account. Both stochastic optimization and robust optimization deal with uncertainties. While a stochastic optimization problem takes into account the distribution of the uncertainty and gives only a probabilistic guarantee of optimal solution, robust optimization hedges against uncertainty with no knowledge of its probability distribution. Another related concept for problems with uncertainties is sensitivity analysis. But for sensitivity analysis, a solution is computed beforehand with nominal data, and then it is checked whether that solution is continuous with respect to small perturbation in the data. Whereas in robust optimization, it is beforehand
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Kuntal Som [email protected] Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
123
Journal of Global Optimization
assumed that the data are uncertain, and then the best possible solution is explored under the uncertainty. The concepts of robust solutions are mainly application-driven, and therefore many different robustness definitions have been proposed by various researchers, for example, min–max robustness, optimistic robustness, regret robustness, light robustness, highly robustness, flimsily robustness, adjustable robustness, etc. to name a few. See [3,8,22] for an overview. Robustness for multi-objective optimization
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