Relative Pareto minimizers for multiobjective problems: existence and optimality conditions

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/...

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Bibliographic Details
Published in:Mathematical programming Vol. 122; no. 2; pp. 301 - 347
Main Authors: Bao, Truong Q., Mordukhovich, Boris S.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.04.2010
Springer
Springer Nature B.V
Subjects:
Applied sciences
Banach spaces
Calculus of variations and optimal control
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Decision theory. Utility theory
Exact sciences and technology
Full Length Paper
Hierarchies
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Operational research and scientific management
Operational research. Management science
Optimization
Optimization techniques
Pareto optimum
Sciences and techniques of general use
Studies
Theoretical
Variational analysis
Necessary optimality conditions
49J53
Existence theorems
49J52
Multiobjective optimization
Variational and extremal principles
Relative Pareto minimizers
90C29
Generalized differentiation
Multivalued function
Subdifferential
Multiobjective programming
Partial ordering
Convex programming
Ordered space
Extremum problem
Nonsmooth analysis
Variational calculus
Infinite dimension
Vector optimization
Mathematical programming
Existence condition
Existence theorem
Pareto optimum
Variational principle
Optimality condition
Optimality criterion
ISSN:0025-5610, 1436-4646
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Summary: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-008-0249-2