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Stronger Kuhn-Tucker Type Conditions in Nonsmooth Multiobjective Optimization: Locally Lipschitz Case

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Abstract

For an inequality constrained nonsmooth multiobjective optimization problem, where the objective and constraint functions are locally Lipschitz, a nonsmooth analogue of the Maeda-type Guignard constraint qualification is given; stronger Kuhn-Tucker type necessary optimality conditions are derived that are expressed in terms of upper convexificators. Moreover, other constraint qualifications sufficient for the nonsmooth analogue are introduced and their relationships are presented.

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References

  1. T. Maeda (1994) ArticleTitleConstraint Qualifications in Multiobjective Optimization Problems: Differentiable Case Journal of Optimization Theory and Applications 80 483–500 Occurrence Handle10.1007/BF02207776 Occurrence Handle0797.90083 Occurrence Handle95b:90105

    Article  MATH  MathSciNet  Google Scholar 

  2. V. Preda I. Chitescu (1999) ArticleTitleOn Constraint Qualifications in Multiobjective Optimization Problems: Semidifferentiable Case Journal of Optimization Theory and Applications 100 417–433 Occurrence Handle10.1023/A:1021794505701 Occurrence Handle99m:90125

    Article  MathSciNet  Google Scholar 

  3. X. F. Li (1994) ArticleTitleConstraint Qualifications in Nonsmooth Multiobjective Optimization Journal of Optimization Theory and Applications 80 483–500 Occurrence Handle10.1007/BF02196594 Occurrence Handle95b:90105

    Article  MathSciNet  Google Scholar 

  4. Demyanov V.F. Convexification and Concavification of Positively Homogeneous Functions by the Same Family of Linear Functions. Technical Report, University of Pisa, pp. 1–11, 1994.

  5. V. F. Demyanov V. Jeyakumar (1997) ArticleTitleHunting for a Smaller Convex Subdifferential Journal of Global Optimization 10 305–326 Occurrence Handle10.1023/A:1008246130864 Occurrence Handle98c:49042

    Article  MathSciNet  Google Scholar 

  6. V. Jeyakumar D. T. Luc (1998) ArticleTitleApproximate Jacobian Matrices for NonSmooth Continuous Maps and C1-Optimization SIAM Journal on Control and Optimization 36 1815–1832 Occurrence Handle10.1137/S0363012996311745 Occurrence Handle99c:49016

    Article  MathSciNet  Google Scholar 

  7. V. Jeyakumar D. T. Luc S. Schaible (1998) ArticleTitleCharacterizations of Generalized Monotone Nonsmooth Continuous Maps Using Approximate Jacobians Journal of Convex Analysis 5 119–132 Occurrence Handle99j:49027

    MathSciNet  Google Scholar 

  8. V. Jeyakumar D. T. Luc (1999) ArticleTitleNonsmooth Calculus, Minimality, and Monotonicity of Convexificators Journal of Optimization Theory and Applications 101 599–621 Occurrence Handle10.1023/A:1021790120780 Occurrence Handle2000e:49023

    Article  MathSciNet  Google Scholar 

  9. V. Jeyakumar W. Wang (1999) ArticleTitleApproximate Hessian Matrices and Second-Order Optimality Conditions for Nonlinear Programming Problems with C1-Data Journal of the Australian Mathematical Society 40 403–420 Occurrence Handle99k:49036

    MathSciNet  Google Scholar 

  10. V. Jeyakumar X. Q. Yang (1999) ArticleTitleApproximate Generalized Hessians and Taylor’s Expansions for Continuously Gateaux Differentiable Functions Nonlinear Analysis: Theory, Methods, and Applications 36 353–368 Occurrence Handle2000f:49018

    MathSciNet  Google Scholar 

  11. X. Wang V. Jeyakumar (2000) ArticleTitleA Sharp Lagrange Multiplier Rule for Nonsmooth Mathematical Programming Problems Involving Equality Constraints SIAM Journal on Optimization 10 1136–1148 Occurrence Handle10.1137/S1052623499354540 Occurrence Handle2001i:90115

    Article  MathSciNet  Google Scholar 

  12. A. Uderzo (2002) Convex Approximators, Convexificators, and Exhausters: Application to Constrained Extremum Problem Kluwer Academic Publication Dordrecht Holland 297–327

    Google Scholar 

  13. J. Dutta S. Chandra (2002) ArticleTitleConvexificators, Generalized Convexity, and Optimality Conditions Journal of Optimization Theory and Application 113 41–64 Occurrence Handle2002m:90097

    MathSciNet  Google Scholar 

  14. D. T. Luc (2002) ArticleTitleA Multiplier Rule for Multiobjective Programming Problems with Continuous Data SIAM Journal on Optimization 13 168–178 Occurrence Handle10.1137/S1052623400378286 Occurrence Handle1055.90063 Occurrence Handle2003d:90086

    Article  MATH  MathSciNet  Google Scholar 

  15. C. Ursescu (1982) ArticleTitleTangent Sets Calculus and Necessary Conditions for Extremality SIAM Journal on Control and Optimization 20 563–574 Occurrence Handle10.1137/0320041 Occurrence Handle0488.49009 Occurrence Handle83j:58018

    Article  MATH  MathSciNet  Google Scholar 

  16. F. H. Clarke (1983) Optimization and Nonsmooth Analysis Wiley-Interscience New York NY

    Google Scholar 

  17. O. L. Mangasarian (1969) Nonlinear Programming McGraw-Hill New York, NY

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Jilin University, Changchun, P. R. China

    X. F. Li

  2. Department of Mathematics, City University of Hong Kong, Hong Kong

    X. F. Li & J. Z. Zhang

Authors
  1. X. F. Li
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  2. J. Z. Zhang
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Additional information

Communicated by H. P. Benson

The authors are grateful to the reviewers for careful reading and helpful comments on the manuscript.

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Li, X.F., Zhang, J.Z. Stronger Kuhn-Tucker Type Conditions in Nonsmooth Multiobjective Optimization: Locally Lipschitz Case. J Optim Theory Appl 127, 367–388 (2005). https://doi.org/10.1007/s10957-005-6550-9

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  • DOI: https://doi.org/10.1007/s10957-005-6550-9

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