Optimality conditions for weak efficiency to vector optimization problems with composed convex functions

Radu Boţ; Ioan Hodrea; Gert Wanka

Open Mathematics (2008)

  • Volume: 6, Issue: 3, page 453-468
  • ISSN: 2391-5455

Abstract

top
We consider a convex optimization problem with a vector valued function as objective function and convex cone inequality constraints. We suppose that each entry of the objective function is the composition of some convex functions. Our aim is to provide necessary and sufficient conditions for the weakly efficient solutions of this vector problem. Moreover, a multiobjective dual treatment is given and weak and strong duality assertions are proved.

How to cite

top

Radu Boţ, Ioan Hodrea, and Gert Wanka. "Optimality conditions for weak efficiency to vector optimization problems with composed convex functions." Open Mathematics 6.3 (2008): 453-468. <http://eudml.org/doc/269647>.

@article{RaduBoţ2008,
abstract = {We consider a convex optimization problem with a vector valued function as objective function and convex cone inequality constraints. We suppose that each entry of the objective function is the composition of some convex functions. Our aim is to provide necessary and sufficient conditions for the weakly efficient solutions of this vector problem. Moreover, a multiobjective dual treatment is given and weak and strong duality assertions are proved.},
author = {Radu Boţ, Ioan Hodrea, Gert Wanka},
journal = {Open Mathematics},
keywords = {multiobjective optimization; composed convex functions; conjugate duality; weak efficiency},
language = {eng},
number = {3},
pages = {453-468},
title = {Optimality conditions for weak efficiency to vector optimization problems with composed convex functions},
url = {http://eudml.org/doc/269647},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Radu Boţ
AU - Ioan Hodrea
AU - Gert Wanka
TI - Optimality conditions for weak efficiency to vector optimization problems with composed convex functions
JO - Open Mathematics
PY - 2008
VL - 6
IS - 3
SP - 453
EP - 468
AB - We consider a convex optimization problem with a vector valued function as objective function and convex cone inequality constraints. We suppose that each entry of the objective function is the composition of some convex functions. Our aim is to provide necessary and sufficient conditions for the weakly efficient solutions of this vector problem. Moreover, a multiobjective dual treatment is given and weak and strong duality assertions are proved.
LA - eng
KW - multiobjective optimization; composed convex functions; conjugate duality; weak efficiency
UR - http://eudml.org/doc/269647
ER -

References

top
  1. [1] Adán M., Novo V., Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness, European J. Oper. Res, 2003, 149, 641–653 http://dx.doi.org/10.1016/S0377-2217(02)00444-7 Zbl1033.90113
  2. [2] Arana-Jiménez M., Rufián-Lizana A., Osuna-Gómez R., Weak efficiency for multiobjective variational problems, European J. Oper. Res., 2004, 155, 373–379 http://dx.doi.org/10.1016/S0377-2217(02)00882-2 Zbl1045.90054
  3. [3] Boţ R.I., Grad S.-M., Wanka G., A new constraint qualification and conjugate duality for composed convex optimization problems, J. Optim. Theory Appl., 2007, 135, 241–255 http://dx.doi.org/10.1007/s10957-007-9247-4 Zbl1146.90494
  4. [4] Boţ R.I., Hodrea I.B., Wanka G., Farkas-type results for inequality systems with composed convex functions via conjugate duality, J. Math. Anal. Appl., 2006, 322, 316–328 http://dx.doi.org/10.1016/j.jmaa.2005.09.007 Zbl1104.90054
  5. [5] Boţ R.I., Kassay G., Wanka G., Strong duality for generalized convex optimization problems, J. Optim. Theory Appl., 2005, 127, 45–70 http://dx.doi.org/10.1007/s10957-005-6392-5 Zbl1158.90420
  6. [6] Boţ R.I., Wanka G., A new duality approach for multiobjective convex optimization problems, J. Nonlinear Convex Anal., 2003, 3, 41–57 Zbl1007.90056
  7. [7] Boţ R.I., Wanka G., An analysis of some dual problems in multiobjective optimization (I), Optimization, 2004, 53, 281–300 http://dx.doi.org/10.1080/02331930410001715514 Zbl1144.90474
  8. [8] Boţ R.I., Wanka G., An analysis of some dual problems in multiobjective optimization (II), Optimization, 2004, 53, 301–324 http://dx.doi.org/10.1080/02331930410001715523 Zbl1144.90474
  9. [9] Boţ R.I., Wanka G., Farkas-type results with conjugate functions, SIAM J. Optim., 2005, 15, 540–554 http://dx.doi.org/10.1137/030602332 Zbl1114.90147
  10. [10] Combari C., Laghdir M., Thibault L., Sous-différentiels de fonctions convexes composées, Ann. Sci. Math. Québec, 1994, 18, 119–148 (in French) 
  11. [11] Hiriart-Urruty J.-B., Martínez-Legaz J.-E., New formulas for the Legendre-Fenchel transform, J. Math. Anal. Appl., 2003, 288, 544–555 http://dx.doi.org/10.1016/j.jmaa.2003.09.012 Zbl1052.49020
  12. [12] Jahn J., Mathematical vector optimization in partially ordered linear spaces, Peter Lang Verlag, Frankfurt am Main, 1986 Zbl0578.90048
  13. [13] Jeyakumar V., Composite nonsmooth programming with Gâteaux differentiability, SIAM J. Optim., 1991, 1, 30–41 http://dx.doi.org/10.1137/0801004 Zbl0752.90067
  14. [14] Jeyakumar V., Lee G.M., Dinh N., Characterizations of solution sets of convex vector minimization problems, European J. Oper. Res., 2006, 174, 1396–1413 http://dx.doi.org/10.1016/j.ejor.2005.05.007 Zbl1103.90090
  15. [15] Jeyakumar V., Yang X.Q., Convex composite minimization with C 1,1 functions, J. Optim. Theory Appl., 1993, 86, 631–648 http://dx.doi.org/10.1007/BF02192162 Zbl0838.90110
  16. [16] Kutateladže S.S., Changes of variables in the Young transformation, Soviet Math. Dokl., 1977, 18, 1039–1041 Zbl0383.49011
  17. [17] Lemaire B., Application of a subdifferential of a convex composite functional to optimal control in variational inequalities, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, 1985, 255, 103–117 Zbl0579.49020
  18. [18] Levin V.L., Sur le Sous-Différentiel de Fonctions Composeé, Doklady Akademia Nauk, 1970, 194, 28–29 (in French) 
  19. [19] Lin Y., Wang X., Necessary and sufficient conditions of optimality for some classical scheduling problems, European J. Oper. Res., 2007, 176, 809–818 http://dx.doi.org/10.1016/j.ejor.2005.09.017 Zbl1103.90043
  20. [20] Rockafellar R.T., Convex analysis, Princeton University Press, Princeton, 1970 Zbl0193.18401
  21. [21] Wanka G., Boţ R.I., Vargyas E., Duality for location problems with unbounded unit balls, European J. Oper. Res., 2007, 179, 1252–1265 http://dx.doi.org/10.1016/j.ejor.2005.09.048 Zbl1127.90047
  22. [22] Yang X.Q., Jeyakumar V., First and second-order optimality conditions for convex composite multiobjective optimization, J. Optim. Theory Appl., 1997, 95, 209–224 http://dx.doi.org/10.1023/A:1022695714596 Zbl0890.90162

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.