• Lee, Gue-Myung ;
  • Kim, Moon-Hee
  • Published : 2003.03.01


Second order necessary optimality condition for properly efficient solutions of a twice differentiable vector optimization problem is given. We obtain a nonsmooth version of the second order necessary optimality condition for properly efficient solutions of a nondifferentiable vector optimization problem. Furthermore, we prove a second order necessary optimality condition for weakly efficient solutions of a nondifferentiable vector optimization problem.


vector optimization problem;properly efficient solutions;second order necessary optimality conditions;second order contingent set;second order adjacent set;singular approximate subdifferential;(Mordukhovich) normal cone


  1. Theory of Games and Statistical Decisions D. Blackwell;M. A. Girshick
  2. Progress in Optimization On Relations between vector variational inequality and vector optimization problem G. M. Lee;X. Q. Yang(et al.)(eds.)
  3. Math. Methods of Oper. Res. v.53 Generalized properly efficient solutions of vector optimization problems D. E. Ward;G. M. Lee
  4. Theoretical Aspects of Industrial Design Sensitivity analysis in nonsmooth optimization B. S. Mordukhovich;D. A. Field(ed.);V. Komkov(ed.)
  5. J. Optim. Theory Appl. v.95 Vector variational inequality and multiobjective pseudolinear programming X. Q. Yang
  6. Optimization and Nonsmooth Analysis F. H. Clarke
  7. Nondifferentiable Optimization V. F. Demyanov and L. V. Vasilev
  8. J. Optim. Theory Appl. v.113 On relations between vector optimization problems and vector variational inequalities D. E. Ward;G. M. Lee
  9. New Trends in Mathematical Programming On Minty variational principle F. Giannessi
  10. J. Math. Anal. Appl. v.154 Second-order necessary conditions for optimality in nonsmooth nonlinear programming M. Studniarski
  11. Theory of Multiobjective Optimization Y. Sawaragi;H. Nakayama;T. Tanino
  12. Optimization v.28 A chain rule for parabolic second-order epiderivatives D. E. Ward
  13. Multiple Criteria Decision Making M. Zeleny
  14. Mathematical Economics T. Takayama
  15. Set-Valued Anal. v.1 Calculus for parabolic second-order derivatives D. E. Ward
  16. Nonsmooth Analysis and Control Theory F. H. Clarke;Yu. S. Ledyaev;R. J. Stern;P. R. Wolenski
  17. J. Math. Anal. Appl. v.71 An improved definition of proper efficiency for vector minimization with respect to cones H. P. Benson
  18. Set-Valued Analysis J. -P. Aubin;H. Frankowska
  19. Mathematical Methods of Game and Economic Theory J. P. Aubin
  20. Mathematical Statistics, A Decision Theoretic Approach T. S. Ferguson
  21. J. Math. Anal. Appl. v.22 Proper efficiency and the theory of vector maximization A. M. Geoffrion
  22. Math. Program. v.67 Optimality conditions in mathematical programming and composite optimization J. P. Penot
  23. Theorems of alternative, quadratic programs and complementarity problems in Variational Inequalities and Complementarity Problems F. Giannessi;R. W. Cottle(ed.);F. Giannessi(ed.);J. L. Lions
  24. Optimization v.31 Epiderivatives of the marginal function in nonsmooth parametric optimization D. E. Ward
  25. Convex Analysis and Nonlinear Optimization J. M. Borwein;A. S. Lewis

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