Kybernetika 53 no. 4, 717-729, 2017

Derivatives of Hadamard type in scalar constrained optimization

Karel PastorDOI: 10.14736/kyb-2017-4-0717

Abstract:

Vsevolod I. Ivanov stated (Nonlinear Analysis 125 (2015), 270-289) the general second-order optimality condition for the constrained vector problem in terms of Hadamard derivatives. We will consider its special case for a scalar problem and show some corollaries for example for ${\ell}$--stable at feasible point functions. Then we show the advantages of obtained results with respect to the previously obtained results.

Keywords:

${\ell }$-stable function, generalized second-order derivative, optimality conditions, $C^{1;1}$-function

Classification:

49K10, 49J52

paper.pdf

References:

  1. D. Bednařík and A. Berková: On some properties of l-stable functions. In: Proc. 14th WSEAS Inter. Conf. on Math. Methods, Comput. Techniques and Intell. Systems, (A J. Viamonte, ed.), Porto 2012.   CrossRef
  2. D. Bednařík and K. Pastor: On second-order conditions in unconstrained optimization. Math. Program. (Ser. A) 113 (2008), 283-298.   DOI:10.1007/s10107-007-0094-8
  3. D. Bednařík and K. Pastor: Differentiability properties of functions that are ${\ell}$-stable at a point. Nonlinear Anal. 69 (2008), 3128-3135.   DOI:10.1016/j.na.2007.09.006
  4. D. Bednařík and K. Pastor: $\ell-$stable functions are continuous. Nonlinear Anal. 70 (2009), 2317-2324.   DOI:10.1016/j.na.2008.03.011
  5. D. Bednařík and K. Pastor: Fréchet approach in second-order optimization. App. Math. Lett. 22 (2009), 960-967.   DOI:10.1016/j.aml.2009.01.019
  6. D. Bednařík and K. Pastor: A note on second-order optimality conditions. Nonlinear Anal. 71 (2009), 1964-1969.   DOI:10.1016/j.na.2009.01.035
  7. D. Bednařík and K. Pastor: ${\ell}$-stable functions are continuous. Nonlinear Anal. 70 (2009), 2317-2324.   DOI:10.1016/j.na.2008.03.011
  8. D. Bednařík and K. Pastor: On l-stable mappings with values in infinite dimensional Banach spaces. Nonlinear Anal. 72 (2010), 1198-1209.   DOI:10.1016/j.na.2009.08.004
  9. D. Bednařík and K. Pastor: On second-order condition in constrained vector optimization. Nonlinear Anal. 74 (2011), 1372-1382.   DOI:10.1016/j.na.2010.10.009
  10. R. Cominetti and R. Correa: A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim. 28 (1990), 789-809.   DOI:10.1137/0328045
  11. W. L. Chan, L. R. Huang and K. F. Ng: On generalized second-order derivatives and Taylor expansions in nonsmooth optimization. SIAM J. Control Optim. 32 (1994), 591-611.   DOI:10.1137/s0363012992227423
  12. M. Dvorská and K. Pastor: On comparison of $\ell-$stable vector optimization results. Math. Slovaca 64 (2014), 4, 1-18.   DOI:10.2478/s12175-014-0252-4
  13. M. Dvorská and K. Pastor K.: Generalization of $C^{1,1}$ property in infinite dimension. Acta Math. Vietnam. 41 (2016), 265-275.   DOI:10.1007/s40306-015-0129-9
  14. V. F. Demyanov and M. Rubinov: Constructive Nonsmooth Analysis Peter Lang, Frankfurt am Main 1995.   CrossRef
  15. I. Ginchev: On scalar and vector $\ell-$stable functions. Nonlinear Anal. 74 (2011), 182-194.   DOI:10.1016/j.na.2010.08.032
  16. I. Ginchev and A. Guerraggio: Second-order conditions for constrained vector optimization problems with ${\ell}$-stable data. Optimization 60 (2011), 179-199.   DOI:10.1080/02331930903578718
  17. I. Ginchev, A. Guerraggio and M. Rocca: From scalar to vector optimization. Appl. Math. 51 (2006), 5-36.   DOI:10.1007/s10492-006-0002-1
  18. A. Guerragio and D. T. Luc: Optimality conditions for $C^{1,1}$ vector optimization problems. J. Optim. Theory Appl. 109 (2001), 615-629.   DOI:10.1023/a:1017519922669
  19. P. G. Georgiev and N. P. Zlateva: Second-order subdifferentials of $C^{1,1}$ functions and optimality conditions. Set-Valued Anal. 4 (1996), 101-117.   DOI:10.1007/bf00425960
  20. J. B. Hiriart-Urruty, J. J. Strodiot and V. H. Nguyen: Generalized Hessian matrix and second-order optimality conditions for problems with $C^{1,1}$ data. Appl. Math. Optim. 11 (1984), 43-56.   DOI:10.1007/bf01442169
  21. V. I. Ivanov: Second-optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization. Nonlinear Anal. 125 (2015), 270-289.   DOI:10.1016/j.na.2015.05.030
  22. J. Jahn: Vector Optimization. Springer-Verlag, New York 2004.   DOI:10.1007/978-3-540-24828-6
  23. V. Jeyakumar and D. T. Luc: Nonsmooth Vector Functions And Continuous Optimization. Springer, Berlin 2008.   CrossRef
  24. P. Q. Khanh and N. D. Tuan: Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming I: ${\ell}$-stability and set-valued directional derivatives. J. Math. Anal. Appl. 403 (2013), 695-702.   DOI:10.1016/j.jmaa.2012.12.076
  25. P. Q. Khanh and N. D. Tuan: Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming II: Optimality conditions. J. Math. Anal. Appl. 403 (2013), 703-714.   DOI:10.1016/j.jmaa.2012.12.075
  26. L. Liu and M. Křížek: The second-order optimality conditions for nonlinear mathematical programming with $C^{1,1}$ data. Appl. Math. 42 (1997), 311-320.   DOI:10.1023/a:1023068513188
  27. L. Liu, P. Neittaanmäki and M. Křížek: Second-order optimality conditions for nondominated solutions of multiobjective programming with $C^{1,1}$ data. Appl. Math. 45 (2000), 381-397.   DOI:10.1023/a:1022272728208
  28. S. J. Li and S. Xu: Sufficient conditions of isolated minimizers for constrained programming problems. Numer. Func. Anal. Optim. 31 (2010), 715-727.   DOI:10.1080/01630563.2010.490970
  29. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer-Verlag, Berlin 2006.   CrossRef
  30. K. Pastor: A note on second-order optimality conditions. Nonlinear Anal. 71 (2009), 1964-1969.   DOI:10.1016/j.na.2009.01.035
  31. K. Pastor: Differentiability properties of ${\ell}$-stable vector functions in infinite-dimensional normed spaces. Taiwanese J. Math. 18 (2014), 187-197.   DOI:10.11650/tjm.18.2014.2605
  32. R. T. Rockafellar: Convex analysis. Princeton University Press, Princeton 1970.   CrossRef
  33. R. T. Rockafellar and R. J.-B. Wets: Variational Analysis. Springer-Verlag, New York 1998.   DOI:10.1007/978-3-642-02431-3
  34. D. L. Torre and M. Rocca: Remarks on second order generalized derivatives for differentiable functions with Lipschitzian jacobian. Appl. Math. E-Notes 3 (2003), 130-137.   CrossRef
  35. X. Q. Yang: On second-order directional derivatives. Nonlinear Anal. 26 (1996), 55-66.   DOI:10.1016/0362-546x(94)00209-z
  36. X. Q. Yang: On relations and applications of generalized second-order directional derivatives. Nonlinear Anal. 36 (1999), 595-614.   DOI:10.1016/s0362-546x(98)00174-6