Scalarization of Levitin–Polyak well-posed set optimization problems

Abstract

The aim of this paper is to study Levitin–Polyak (LP in short) well-posedness for set optimization problems. We define the global notions of metrically well-setness and metrically LP well-setness and the pointwise notions of LP well-posedness, strongly DH-well-posedness and strongly B-well-posedness for set optimization problems. Using a scalarization function defined by means of the point-to-set distance, we characterize the LP well-posedness and the metrically well-setness of a set optimization problem through the LP well-posedness and the metrically well-setness of a scalar optimization problem, respectively.

Keywords:

• Levitin–Polyak well-posedness
• metrically well-setness
• set optimization
• scalarization functions
• point-to-set distance

AMS Subject Classifications:

• 80M50
• 49K40
• 54C60

Acknowledgements

We would like to express our gratitude to the anonymous referees for their helpful comments on the first version of this paper.

Notes

No potential conflict of interest was reported by the authors.