On some efficiency conditions for vector optimization problems with uncertain cone constraints: a robust approach via set-valued inclusions

Abstract

In the present paper, several types of efficiency conditions are established for vector optimization problems with cone constraints affected by uncertainty, but with no information of stochastic nature about the uncertain data. Following a robust optimization approach, data uncertainty is faced by handling set-valued inclusion problems. The employment of recent advances about error bounds and tangential approximations of the solution set to the latter enables one to achieve necessary conditions for weak efficiency via a penalization method as well as via the modern revisitation of the Euler–Lagrange method, with or without generalized convexity assumptions. The presented conditions are formulated in terms of various nonsmooth analysis constructions, expressing first-order approximations of mappings and sets, while the metric increase property plays the role of a constraint qualification.

Keywords:

• Vector optimization problem
• data uncertainty
• robust approach
• weak efficiency condition
• generalized derivative

Acknowledgments

The author is grateful to both the anonymous referees for their useful remarks.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 To be more precise, in [9], the authors do indicate as a forerunner of their approach A.L. Soyster, who in [34] introduced a similar point of view, in dealing with uncertainly constrained problems in mathematical programming.