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Stability and Scalarization in Vector Optimization Using Improvement Sets

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Abstract

The aim of this paper is to study certain aspects of stability and scalarization of a nonconvex vector optimization problem through improvement sets. This paper attempts to investigate an open problem on stability posed by Chicco et al. The notion of stability is studied through Painlevé–Kuratowski set-convergence, where we establish sufficiency conditions for the lower and upper set-convergences of optimal solution sets of a family of perturbed vector problems, both in the given space and its image space. The perturbations are performed both on the objective function and the feasible set. Further, by using a nonlinear scalarization function defined in terms of an improvement set, we establish lower and upper Painlevé–Kuratowski set-convergences of sequences of approximate solution sets of certain scalarized problems. We then link these set-convergences with the set-convergences of optimal solution sets of the perturbed problems. Finally, we investigate the stability and scalarization of a linear vector optimization problem in finite dimensional spaces.

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Acknowledgments

The authors are grateful to Prof. F. Giannessi for his valuable comments and suggestions which helped in improving the paper and led to Sect. 5 of the paper. Research of C.S. Lalitha is supported by R&D Doctoral Research Programme for University Faculty.

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Authors and Affiliations

  1. Department of Mathematics, University of Delhi South Campus, Benito Jaurez Road, New Delhi, 110021, India

    C. S. Lalitha

  2. Department of Mathematics, St. Stephen’s College, University of Delhi, Delhi, 110007, India

    Prashanto Chatterjee

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  1. C. S. Lalitha
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  2. Prashanto Chatterjee
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Correspondence to Prashanto Chatterjee.

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Lalitha, C.S., Chatterjee, P. Stability and Scalarization in Vector Optimization Using Improvement Sets. J Optim Theory Appl 166, 825–843 (2015). https://doi.org/10.1007/s10957-014-0686-4

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  • DOI: https://doi.org/10.1007/s10957-014-0686-4

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