Decrease of C1,1 property in vector optimization

Dušan Bednařík; Karel Pastor

RAIRO - Operations Research (2009)

  • Volume: 43, Issue: 4, page 359-372
  • ISSN: 0399-0559

Abstract

top
In the paper we generalize sufficient and necessary optimality conditions obtained by Ginchev, Guerraggio, Rocca, and by authors with the help of the notion of ℓ-stability for vector functions.

How to cite

top

Bednařík, Dušan, and Pastor, Karel. "Decrease of C1,1 property in vector optimization." RAIRO - Operations Research 43.4 (2009): 359-372. <http://eudml.org/doc/250629>.

@article{Bednařík2009,
abstract = { In the paper we generalize sufficient and necessary optimality conditions obtained by Ginchev, Guerraggio, Rocca, and by authors with the help of the notion of ℓ-stability for vector functions. },
author = {Bednařík, Dušan, Pastor, Karel},
journal = {RAIRO - Operations Research},
keywords = {C1,1 function; ℓ-stable function; generalized second-order directional derivative; Dini derivative; weakly efficient minimizer; isolated minimizer of second-order.; function; -stable function; generalized second-order directional derivative; isolated minimizer of second-order},
language = {eng},
month = {10},
number = {4},
pages = {359-372},
publisher = {EDP Sciences},
title = {Decrease of C1,1 property in vector optimization},
url = {http://eudml.org/doc/250629},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Bednařík, Dušan
AU - Pastor, Karel
TI - Decrease of C1,1 property in vector optimization
JO - RAIRO - Operations Research
DA - 2009/10//
PB - EDP Sciences
VL - 43
IS - 4
SP - 359
EP - 372
AB - In the paper we generalize sufficient and necessary optimality conditions obtained by Ginchev, Guerraggio, Rocca, and by authors with the help of the notion of ℓ-stability for vector functions.
LA - eng
KW - C1,1 function; ℓ-stable function; generalized second-order directional derivative; Dini derivative; weakly efficient minimizer; isolated minimizer of second-order.; function; -stable function; generalized second-order directional derivative; isolated minimizer of second-order
UR - http://eudml.org/doc/250629
ER -

References

top
  1. J.P. Aubin and A. Cellina, Differential inclusions, Springer Verlag, Berlin (1984).  
  2. D. Bednařík and K. Pastor, On characterization of convexity for regularly locally Lipschitz functions. Nonlinear Anal.57 (2004) 85–97.  Zbl1073.49008
  3. D. Bednařík and K. Pastor, Elimination of strict convergence in optimization. SIAM J. Control Optim.43 (2004) 1063–1077.  Zbl1089.49023
  4. D. Bednařík and K. Pastor, Using the Peano derivative in unconstrained optimization. Math. Program.113 (2008) 283–298.  Zbl1211.90276
  5. D. Bednařík and K. Pastor, Differentiability properties of functions that are ℓ-stable at a point. Nonlinear Anal.69 (2008) 3128–3135.  Zbl1146.49017
  6. D. Bednařík and K. Pastor, ℓ-stable functions are continuous. Nonlinear Anal.70 (2009) 2317–2324.  Zbl1158.49022
  7. A. Ben-tal and J. Zowe, Directional derivatives in nonsmooth optimization. J. Optim. Theory Appl.47 (1985) 483–490.  Zbl0556.90074
  8. 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.  Zbl0801.49016
  9. R. Cominetti and R. Correa, A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim.28 (1990) 789–809.  Zbl0714.49020
  10. P.G. Georgiev and N.P. Zlateva, Second-order Subdifferentials of C1,1 Functions and Optimality Conditions. Set-Valued Anal.4 (1996) 101–117.  Zbl0864.49012
  11. I. Ginchev, Higher order optimality conditions in nonsmooth optimization. Optimization51 (2002) 47–72.  Zbl1011.49014
  12. I. Ginchev, A. Guerraggio and M. Rocca, Second order conditions for C1,1 constrained vector optimization. Math. Program. Ser. B104 (2005) 389–405.  Zbl1102.90058
  13. I. Ginchev, A. Guerraggio and M. Rocca, From scalar to vector optimization. Applications of Mathematics51 (2006) 5–36.  Zbl1164.90399
  14. A. Guerraggio and D.T. Luc, Optimality conditions for C1,1 vector optimization problems. J. Optim. Theory Appl.109 (2001) 615–629.  Zbl1038.49027
  15. J.J. Hiriart–Urruty, J.J. Strodiot and V.H. Nguyen, Generalized Hessian matrix and second order optimality conditions for problems with C1,1 data. Appl. Math. Optim.11 (1984) 169–180.  Zbl0542.49011
  16. L.R. Huang and K.F. Ng, On lower bounds of the second-order directional derivatives of Ben-Tal-Zowe and Chaney. Math. Oper. Res.22 (1997) 747–753.  Zbl0886.49018
  17. J. Jahn, Vector optimization, Springer Verlag, New York (2004).  Zbl1055.90065
  18. B. Jiménez and V. Novo, First and second order sufficient conditions for strict minimality in nonsmooth vector optimization. J. Math. Anal. Appl.284 (2003) 496–510.  Zbl1033.90120
  19. B. Jiménez and V. Novo, First order optimality conditions in vector optimization involving stable functions. Optimization57 (2008) 449–471.  Zbl1191.90056
  20. P.Q. Khanh and N.D. Tuan, Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. J. Optim. Theory Appl.133 (2007) 341–357.  Zbl1149.90134
  21. D. Klatte, Upper Lipschitz behavior of solutions to perturbed C1,1 programs. Math. Program. (Ser B)88 (2000) 285–311.  Zbl1017.90111
  22. L. Liu, The second-order conditions of nondominated solutions for C1,1 generalized multiobjective mathematical programming. J. Systems Sci. Math. Sci.4 (1991) 128–138.  Zbl0734.90078
  23. L. Liu and M. Křížek, The second-order optimality conditions for nonlinear mathematical programming with C1,1 data. Appl. Math.42 (1997) 311–320.  Zbl0903.90152
  24. L. Liu, P. NeittaanmÄki and M. Křížek, Second-order optimality conditions for nondominated solutions of multiobjective programming with C1,1 data. Appl. Math.45 (2000) 381–397.  Zbl0995.90085
  25. Y. Maruyama, Second-order necessary conditions for nonlinear optimization problems in Banach spaces and their application to an optimal control problem. Math. Oper. Res.15 (1990) 467–482.  Zbl0718.49024
  26. K. Pastor, Convexity and generalized second-order derivatives for locally Lipschitz functions. Nonlinear Anal.60 (2005) 547–555.  Zbl1067.49014
  27. K. Pastor, Fréchet approach to generalized second-order differentiability. to appear in Studia Scientiarum Mathematicarum Hungarica45 (2008) 333–352.  Zbl1199.49060
  28. R.T. Rockafellar, Convex analysis, Princeton University Press, Princeton (1970).  Zbl0193.18401
  29. R.T. Rockafellar, R.J.-B. Wets, Variational Analysis, Springer Verlag, New York (1998).  Zbl0888.49001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.