Abstract
In this paper we derive first order necessary and sufficient optimality conditions for nonsmooth optimization problems with multiple criteria. These conditions are given for different optimality notions (i.e. weak, Pareto- and proper minimality) and for different types of derivatives of nonsmooth objective functions (locally Lipschitz continuous and quasidifferentiable) mappings. The conditions are given, if possible, in terms of a derivative and a subdifferential of those mappings.
Similar content being viewed by others
References
Borwein JM (1977) Proper Efficient Points for Maximization with Respect to Cones. SIAM J Control Optim, vol 15
Borwein JM (1978) Weak Tangent Cones and Optimization in a Banach Space. SIAM J Control Optim 16:512–522
Borwein JM (1982) Continuity and Differentiability Properties of Convex Operators. Proc London Math Soc 44:420–444
Borwein JM (1980) The Geometry of PARETO Efficiency over Cones. Math Operationsforsch Statist, Ser Optim 11:235–248
Choo EU, Atkins DR (1983) Proper Efficiency in Nonconvex Multicriteria Programming. Math Oper Res vol 8
Clarke FH (1973) Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations. Ph D Thesis, Univ of Washington
Clarke FH (1983) Optimization and Nonsmooth Analysis. Wiley Interscience, New York
Demjanov VF, Rubinov AM (1983) On Quasidifferentiable Mappings. Math Operationsforsch Statist, Ser Optim 14:3–21
Jahn J (1986) Mathematical Vector Optimization in Partially Ordered Linear Spaces. Verlag Peter Lang, Frankfurt/Main
Jahn J (1987) Parametric Approximation Problems Arising in Vector Optimization. J Optim Theory Appl 54:503–516
Jahn J, Sachs E (1986) Generalized Quasiconvex Mappings and Vector Optimization. SIAM J Control Optim, vol 24
Kaliszewski I (1985) A Characterization of Properly Efficient Solutions by an Augmented Tchebycheff Norm. Bull Polish Acad of Sciences, vol 33, no 7–8
Minami W (1983) Weak Pareto-Optimal Necessary Conditions in Nondifferentiable Multiobjective Program on a Banach Space. J Optim Theory Appl 41:451–461
Staib T (1989) Notwendige Optimalitätsbedingungen in der mehrkriteriellen Optimierung mit Anwendung auf Steuerungsprobleme. Doctoral Dissertation, University of Erlangen-Nürnberg
Valadier M (1972) Sous-Différentiabilité de Fonctions Convexes á Valeurs dans Un Espace Vectoriel Ordonné. Math Scand 30–5
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Staib, T. On necessary and sufficient optimality conditions for multicriterial optimization problems. ZOR - Methods and Models of Operations Research 35, 231–248 (1991). https://doi.org/10.1007/BF01415909
Issue Date:
DOI: https://doi.org/10.1007/BF01415909