Vector Optimization

The studies in this area are quite new for this group and have been the topic of a recently discussed Ph.D. dissertation [B03].
A deep analysis of Lagrangian regularity has been developed: relying on the separation properties investigated in [BP98], a classification of vector optimization problems based on Lagrangian multipliers has been proposed in [BP99a]; it allows not only to achieve very weak regularity conditions but also those ones, which guarantee additional properties of multipliers such as boundededness of multipliers [BP00] and uniqueness [BC00a], for a wide class of differentiable and nondifferentiable problems. Applications to both scalar and vector saddlepoints of Lagrangians related to vector optimization problems have been developed in [BP99b], [B02], leading to new and stronger optimality criteria.
Optimality criteria based on second order analysis have been investigated in [BC00], [B03a/b] for smooth vector optimization problems.
A set-valued Dini derivative for vector-valued functions has been introduced in [BP00], considering suitable limits of vector incremental ratios without relying on components; it has been employed to achieve new necessary optimality conditions for nondifferentiable vector problems. The analysis of this kind of approach has been deepen and extended to second order analysis in [B03a], showing also how to recover well-known tools for vector functions such as generalized Jacobians and Hessians as particular cases; furthermore, a comparison with the standard componentwise approach has shown that this approach provides sharper results [B04]. Finally, general differentiability concepts for set-valued maps and the corresponding optimality criteria have been investigated in [BC02].

[B03b] G. Bigi "On Sufficient Second Order Optimality Conditions in Multiobjective Optimization" Mathematical Methods of Operations Research, to appear.

[BP03] G. Bigi and M. Pappalardo "About the Duality Gap in Vector Optimization" in Variational Analysis A. Maugeri editor, Kluwer, to appear (revised version of the Technical Report 01-15, Dipartimento di Informatica, Università di Pisa).

[[B04] G. Bigi "Componentwise versus Global Approaches to Nonsmooth Multiobjective Optimization" Journal of Industrial and Management Optimization, 1, 21-32, 2005.

BC00a] G. Bigi and M. Castellani "Uniqueness of KKT Multipliers in Multiobjective Programming" Applied Mathematics Letters, 17, 1985-1290, 2004.

[B03a] G. Bigi "Optimality and Lagrangian Regularity in Vector Optimization" Ph.D. Thesis in Mathematics, Università di Pisa, S.E.U., pp. 144, 2003.

[BC02] G. Bigi and M. Castellani "K-epiderivatives for Set-Valued Functions and Optimization" Mathematical Methods of Operations Research, 55, 401-412, 2002.

[B02] G. Bigi "Saddlepoint Optimalty Criteria in Vector Optimization" in Optimization in Economics, Finance and Industry, A. Guerraggio et al. editors, Datanova, 85-102, 2002.

[BC00b] G. Bigi and M. Castellani "Second Order Optimality Conditions for Differentiable Multiobjective Problems" Rairo-Operations Research, 34, 411-426, 2000.

[BP00] G. Bigi and M. Pappalardo "Generalized Lagrange Multipliers: Regularity and Boundedness" in Nonlinear Optimization and Related Topics, G. Di Pillo and F. Giannessi editors, Kluwer, 1-14, 2000.

[BP99a] G. Bigi and M. Pappalardo "Regularity Conditions in Vector Optimization" Journal of Optimization Theory and Applications 102, 83-96, 1999.

[BP99b] G. Bigi and M. Pappalardo "On Lagrangian Saddlepoints in Vector Optimization" in Generalized Convexity and Optimization for Economic and Financial Decisions, G. Giorgi and F. Rossi editors, Pitagora, pp. 33-46, 1999.

[BP98] G. Bigi and M. Pappalardo "Regularity Conditions for The Linear Separation of Sets" Journal of Optimization Theory and Applications 99, 533-540, 1998.