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Second-Order Optimality Conditions in Set Optimization

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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

    Google Scholar 

  • J.P. Aubin H. Frankowska (1990) Set Valued Analysis Birkhäuser Boston, Massachusetts

    Google Scholar 

  • J. Jahn R. Rauh (1997) ArticleTitleContingent Epiderivatives and Set-Valued Optimization Mathematical Methods of Operations Research 46 193–211

    Google Scholar 

  • E.M. Bednarczuk W. Song (1998) ArticleTitleContingent Epiderivative and Its Application to Set-Valued Optimization Control and Cybernetics 27 375–386

    Google Scholar 

  • G.Y. Cheng J. Jahn (1998) ArticleTitleOptimality Conditions for Set-Valued Optimization Problems Mathematical Methods of Operations Research 48 187–200

    Google Scholar 

  • J.F. Bonnans A. Shapiro (2000) Perturbation Analysis of Optimization Problems Springer Series in Operations Research New York NY

    Google Scholar 

  • 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

    Google Scholar 

  • M. Castellani M. Pappalardo (1996) ArticleTitleLocal Second Order Approximations and Applications in Optimization Optimization 37 305–321

    Google Scholar 

  • B. Jiménez V. Novo (2003) ArticleTitleSecond Order Necessary Conditions in Set Constrained Differentiable Vector Optimization Mathematical Methods of Operations Research 58 299–317

    Google Scholar 

  • J.P. Penot (1999) ArticleTitleSecond-Order Conditions for Optimization Problems with Constraints SIAM Journal on Optimization 37 303–318

    Google Scholar 

  • D. Ward (1993) ArticleTitleCalculus for Parabolic Second-Order Derivatives Set-Valued Analysis. 1 213–246

    Google Scholar 

  • J. Jahn (2004) Vector Optimization: Theory Applications and Extensions Springer Berlin Germany

    Google Scholar 

  • 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

    Google Scholar 

  • G. Bigi M. Castellani (2000) ArticleTitleSecond Order Optimality Conditions for Differentiable Multiobjective Problems RAIRO Operations Research 34 411–426

    Google Scholar 

  • A. Götz J. Jahn (1999) ArticleTitleThe Lagrange Multiplier Rule in Set-Valued Optimization SIAM Journal on Optimization 10 331–344

    Google Scholar 

  • J. Jahn (1996) Introduction to the Theory of Nonlinear Optimization Springer Berlin Germany

    Google Scholar 

  • J. Baier J. Jahn (1999) ArticleTitleOn Subgradients of Set-Valued Maps Journal of Optimization Theory and Applications 100 233–240

    Google Scholar 

  • 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

    Google Scholar 

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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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