Abstract
In this paper, we propose second-order epiderivatives for set-valued maps. By using these concepts, second-order necessary optimality conditions and a sufficient optimality condition are given in set optimization. These conditions extend some known results in optimization.
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References
H.W. Corley (1998) ArticleTitleOptimality Conditions for Maximization of Set-Valued Functions Journal of Optimization Theory and Applications 58 1–10
J.P. Aubin H. Frankowska (1990) Set Valued Analysis Birkhäuser Boston, Massachusetts
J. Jahn R. Rauh (1997) ArticleTitleContingent Epiderivatives and Set-Valued Optimization Mathematical Methods of Operations Research 46 193–211
E.M. Bednarczuk W. Song (1998) ArticleTitleContingent Epiderivative and Its Application to Set-Valued Optimization Control and Cybernetics 27 375–386
G.Y. Cheng J. Jahn (1998) ArticleTitleOptimality Conditions for Set-Valued Optimization Problems Mathematical Methods of Operations Research 48 187–200
J.F. Bonnans A. Shapiro (2000) Perturbation Analysis of Optimization Problems Springer Series in Operations Research New York NY
Cambini A. and Martein L. First and Second Order Optimality Conditions in Vector Optimization Preprint Department of Statistics and Applied Mathematics University of Pisa 2000.
A. Cambini L. Martein M. Vlach (1999) ArticleTitleSecond Order Tangent Sets and Optimality Conditions Mathematica Japonica 49 451–461
M. Castellani M. Pappalardo (1996) ArticleTitleLocal Second Order Approximations and Applications in Optimization Optimization 37 305–321
B. Jiménez V. Novo (2003) ArticleTitleSecond Order Necessary Conditions in Set Constrained Differentiable Vector Optimization Mathematical Methods of Operations Research 58 299–317
J.P. Penot (1999) ArticleTitleSecond-Order Conditions for Optimization Problems with Constraints SIAM Journal on Optimization 37 303–318
D. Ward (1993) ArticleTitleCalculus for Parabolic Second-Order Derivatives Set-Valued Analysis. 1 213–246
J. Jahn (2004) Vector Optimization: Theory Applications and Extensions Springer Berlin Germany
Luc D.T. Theory of Vector Optimization Lecture Notes in Economics and Mathematical Sciences Springer Berlin Germany Vol. 319 1988.
Y. Sonntag C. Zalinescu (2000) ArticleTitleComparison of Existence Results for Efficient Points Journal of Optimization Theory and Applications 105 161–188
G. Bigi M. Castellani (2000) ArticleTitleSecond Order Optimality Conditions for Differentiable Multiobjective Problems RAIRO Operations Research 34 411–426
A. Götz J. Jahn (1999) ArticleTitleThe Lagrange Multiplier Rule in Set-Valued Optimization SIAM Journal on Optimization 10 331–344
J. Jahn (1996) Introduction to the Theory of Nonlinear Optimization Springer Berlin Germany
J. Baier J. Jahn (1999) ArticleTitleOn Subgradients of Set-Valued Maps Journal of Optimization Theory and Applications 100 233–240
Mordukhovich B.S. Sensitivity Analysis in Nonsmooth Optimization Theoretical Aspects of Industrial Design Edited by D. A. Filed and V. Komkov SIAM Proceedings in Applied Mathematics Vol. 58 pp. 32–46 1992.
B.S. Mordukhovich J.V. Outrata (2001) ArticleTitleOn Second-Order Subdifferentials and Their Applications SIAM Journal on Optimization 12 139–169
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Communicated by S. Schaible
The authors are grateful to the referees for careful reading and helpful remarks.
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Jahn, J., Khan, A.A. & Zeilinger, P. Second-Order Optimality Conditions in Set Optimization. J Optim Theory Appl 125, 331–347 (2005). https://doi.org/10.1007/s10957-004-1841-0
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DOI: https://doi.org/10.1007/s10957-004-1841-0