Approximate solutions and scalarization in set-valued optimization

Abstract

Certain notions of approximate weak efficient solutions are considered for a set-valued optimization problem based on vector and set criteria approaches. For approximate solutions based on the vector approach, a characterization is provided in terms of an extended Gerstewitz’s function. For the set approach case, two notions of approximate weak efficient solutions are introduced using a lower and an upper quasi order relations for sets and further compactness and stability aspects are discussed for these approximate solutions. Existence and scalarization using a generalized Gerstewitz’s function are also established for approximate solutions, based on the lower set order relation.

Keywords:

• Set-valued optimization
• approximate solutions
• scalarization
• Gerstewitz function
• Painlevé–Kuratowski convergence

Acknowledgements

The authors would like to thank the anonymous referees for providing many helpful suggestions which improved the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Research of C. S. Lalitha was supported by R&D Doctoral Research Programme Fund for University faculty, Research Council, University of Delhi [grant number RC/2015/9677].