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Mathematical Programming

, Volume 122, Issue 2, pp 301–347 | Cite as

Relative Pareto minimizers for multiobjective problems: existence and optimality conditions

  • Truong Q. BaoEmail author
  • Boris S. Mordukhovich
FULL LENGTH PAPER Series A

Abstract

In this paper we introduce and study enhanced notions of relative Pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers for general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient counterparts) that are new in both finite-dimensional and infinite-dimensional settings. Our proofs are based on variational and extremal principles of variational analysis; in particular, on new versions of the Ekeland variational principle and the subdifferential variational principle for set-valued and single-valued mappings in infinite-dimensional spaces.

Keywords

Multiobjective optimization Variational analysis Relative Pareto minimizers Existence theorems Necessary optimality conditions Variational and extremal principles Generalized differentiation 

Mathematics Subject Classification (2000)

90C29 49J52 49J53 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA

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