Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality

Liana Cioban; Ernö Csetnek

Open Mathematics (2013)

  • Volume: 11, Issue: 5, page 829-850
  • ISSN: 2391-5455

Abstract

top
Based on conjugate duality we construct several gap functions for general variational inequalities and equilibrium problems, in the formulation of which a so-called perturbation function is used. These functions are written with the help of the Fenchel-Moreau conjugate of the functions involved. In case we are working in the convex setting and a regularity condition is fulfilled, these functions become gap functions. The techniques used are the ones considered in [Altangerel L., Boţ R.I., Wanka G., On gap functions for equilibrium problems via Fenchel duality, Pac. J. Optim., 2006, 2(3), 667–678] and [Altangerel L., Boţ R.I., Wanka G., On the construction of gap functions for variational inequalities via conjugate duality, Asia-Pac. J. Oper. Res., 2007, 24(3), 353–371]. By particularizing the perturbation function we rediscover several gap functions from the literature. We also characterize the solutions of various variational inequalities and equilibrium problems by means of the properties of the convex subdifferential. In case no regularity condition is fulfilled, we deliver also necessary and sufficient sequential characterizations for these solutions. Several examples are illustrating the theoretical aspects.

How to cite

top

Liana Cioban, and Ernö Csetnek. "Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality." Open Mathematics 11.5 (2013): 829-850. <http://eudml.org/doc/269363>.

@article{LianaCioban2013,
abstract = {Based on conjugate duality we construct several gap functions for general variational inequalities and equilibrium problems, in the formulation of which a so-called perturbation function is used. These functions are written with the help of the Fenchel-Moreau conjugate of the functions involved. In case we are working in the convex setting and a regularity condition is fulfilled, these functions become gap functions. The techniques used are the ones considered in [Altangerel L., Boţ R.I., Wanka G., On gap functions for equilibrium problems via Fenchel duality, Pac. J. Optim., 2006, 2(3), 667–678] and [Altangerel L., Boţ R.I., Wanka G., On the construction of gap functions for variational inequalities via conjugate duality, Asia-Pac. J. Oper. Res., 2007, 24(3), 353–371]. By particularizing the perturbation function we rediscover several gap functions from the literature. We also characterize the solutions of various variational inequalities and equilibrium problems by means of the properties of the convex subdifferential. In case no regularity condition is fulfilled, we deliver also necessary and sufficient sequential characterizations for these solutions. Several examples are illustrating the theoretical aspects.},
author = {Liana Cioban, Ernö Csetnek},
journal = {Open Mathematics},
keywords = {Variational inequalities; Gap functions; Dual gap functions; Equilibrium problems; Perturbation theory; Sequential optimality conditions; variational inequalities; gap functions; dual gap functions; equilibrium problems; perturbation theory; sequential optimality conditions},
language = {eng},
number = {5},
pages = {829-850},
title = {Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality},
url = {http://eudml.org/doc/269363},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Liana Cioban
AU - Ernö Csetnek
TI - Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality
JO - Open Mathematics
PY - 2013
VL - 11
IS - 5
SP - 829
EP - 850
AB - Based on conjugate duality we construct several gap functions for general variational inequalities and equilibrium problems, in the formulation of which a so-called perturbation function is used. These functions are written with the help of the Fenchel-Moreau conjugate of the functions involved. In case we are working in the convex setting and a regularity condition is fulfilled, these functions become gap functions. The techniques used are the ones considered in [Altangerel L., Boţ R.I., Wanka G., On gap functions for equilibrium problems via Fenchel duality, Pac. J. Optim., 2006, 2(3), 667–678] and [Altangerel L., Boţ R.I., Wanka G., On the construction of gap functions for variational inequalities via conjugate duality, Asia-Pac. J. Oper. Res., 2007, 24(3), 353–371]. By particularizing the perturbation function we rediscover several gap functions from the literature. We also characterize the solutions of various variational inequalities and equilibrium problems by means of the properties of the convex subdifferential. In case no regularity condition is fulfilled, we deliver also necessary and sufficient sequential characterizations for these solutions. Several examples are illustrating the theoretical aspects.
LA - eng
KW - Variational inequalities; Gap functions; Dual gap functions; Equilibrium problems; Perturbation theory; Sequential optimality conditions; variational inequalities; gap functions; dual gap functions; equilibrium problems; perturbation theory; sequential optimality conditions
UR - http://eudml.org/doc/269363
ER -

References

top
  1. [1] Altangerel L., Boţ R.I., Wanka G., On gap functions for equilibrium problems via Fenchel duality, Pac. J. Optim., 2006, 2(3), 667–678 Zbl1103.49016
  2. [2] Altangerel L., Boţ R.I., Wanka G., On the construction of gap functions for variational inequalities via conjugate duality, Asia-Pac. J. Oper. Res., 2007, 24(3), 353–371 http://dx.doi.org/10.1142/S0217595907001309 Zbl1141.49303
  3. [3] Auslender A., Optimisation, Masson, Paris-New York-Barcelona, 1976 
  4. [4] Aussel D., Hadjisavvas N., On quasimonotone variational inequalities, J. Optim. Theory Appl., 2004, 121(2), 445–450 http://dx.doi.org/10.1023/B:JOTA.0000037413.45495.00 Zbl1062.49006
  5. [5] Bauschke H.H., Combettes P.L., Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books Math./Ouvrages Math. SMC, Springer, New York, 2011 Zbl1218.47001
  6. [6] Blum E., Oettli W., From optimization and variational inequalities to equilibrium problems, Math. Student, 1994, 63(1–4), 123–145 Zbl0888.49007
  7. [7] Boţ R.I., Conjugate Duality in Convex Optimization, Lecture Notes in Econom. and Math. Systems, 637, Springer, Berlin, 2010 Zbl1190.90002
  8. [8] Boţ R.I., Capătă A.E., Existence results and gap functions for the generalized equilibrium problem with composed functions, Nonlinear Anal., 2010, 72(1), 316–324 http://dx.doi.org/10.1016/j.na.2009.06.055 Zbl1180.49001
  9. [9] Boţ R.I., Csetnek E.R., Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements, Optimization, 2012, 61(1), 35–65 http://dx.doi.org/10.1080/02331934.2010.505649 Zbl1246.46062
  10. [10] Boţ R.I., Csetnek E.R., Wanka G., Sequential optimality conditions in convex programming via perturbation approach, J. Convex Anal., 2008, 15(1), 149–164 Zbl1144.90018
  11. [11] Boţ R.I., Csetnek E.R., Wanka G., Sequential optimality conditions for composed convex optimization problems, J. Math. Anal. Appl., 2008, 342(2), 1015–1025 http://dx.doi.org/10.1016/j.jmaa.2007.12.066 Zbl1220.90087
  12. [12] Boţ R.I., Grad S.-M., Lower semicontinuous type regularity conditions for subdifferential calculus, Optim. Methods Softw., 2010, 25(1), 37–48 http://dx.doi.org/10.1080/10556780903208977 Zbl1220.90158
  13. [13] Boţ R.I., Grad S.-M., Wanka G., Duality in Vector Optimization, Vector Optim., Springer, Berlin, 2009 Zbl1177.90355
  14. [14] Boţ R.I., Wanka G., A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces, Nonlinear Anal., 2006, 64(12), 2787–2804 http://dx.doi.org/10.1016/j.na.2005.09.017 Zbl1087.49026
  15. [15] Burachik R.S., Jeyakumar V., A new geometric condition for Fenchel’s duality in infinite dimensional spaces, Math. Program., 2005, 104(2-3), 229–233 http://dx.doi.org/10.1007/s10107-005-0614-3 Zbl1093.90077
  16. [16] Burachik R.S., Jeyakumar V., Wu Z.-Y., Necessary and sufficient conditions for stable conjugate duality, Nonlinear Anal., 2006, 64(9), 1998–2006 http://dx.doi.org/10.1016/j.na.2005.07.034 Zbl1091.49031
  17. [17] Chen G.Y., Goh C.J., Yang X.Q., On gap functions and duality of variational inequality problems, J. Math. Anal. Appl., 1997, 214(2), 658–673 http://dx.doi.org/10.1006/jmaa.1997.5608 
  18. [18] Cioban L., Csetnek E.R., Duality for ɛ-variational inequalities via the subdifferential calculus, Nonlinear Anal., 2012, 75(6), 3142–3156 http://dx.doi.org/10.1016/j.na.2011.12.012 Zbl1236.58028
  19. [19] Csetnek E.R., Overcoming the Failure of the Classical Generalized Interior-point Regularity Conditions in Convex Optimization. Applications of the Duality Theory to Enlargements of Maximal Monotone Operators, Logos, Berlin, 2010 
  20. [20] Dinh N., Strodiot J.J., Nguyen V.H., Duality and optimality conditions for generalized equilibrium problems involving DC functions, J. Global Optim., 2010, 48(2), 183–208 http://dx.doi.org/10.1007/s10898-009-9486-z Zbl1228.90078
  21. [21] Ekeland I., Temam R., Convex Analysis and Variational Problems, Stud. Math. Appl., 1, North-Holland, Amsterdam-Oxford, 1976 Zbl0322.90046
  22. [22] Facchinei F., Pang J.-S., Finite-Dimensional Variational Inequalities and Complementarity Problems, I, II, Springer Ser. Oper. Res., Springer, New York, 2003 Zbl1062.90002
  23. [23] Giannessi F., On some connections among variational inequalities, combinatorial and continuous optimization, Ann. Oper. Res., 1995, 58, 181–200 http://dx.doi.org/10.1007/BF02032131 Zbl0844.90069
  24. [24] Giannessi F., On Minty variational principle, In: New Trends in Mathematical Programming, Appl. Optim., 13, Kluwer, Boston, 1998, 93–99 Zbl0909.90253
  25. [25] Goh C.J., Yang X.Q., Duality in Optimization and Variational Inequalities, Optim. Theory Appl., 2, Taylor & Francis, London, 2002 Zbl1125.90059
  26. [26] Gowda M.S., Teboulle M., A comparison of constraint qualifications in infinite-dimensional convex programming, SIAM J. Control Optim., 1990, 28(4), 925–935 http://dx.doi.org/10.1137/0328051 Zbl0713.49042
  27. [27] Harker P.T., Pang J.-S., Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming, 1990, 48(2), 161–220 http://dx.doi.org/10.1007/BF01582255 Zbl0734.90098
  28. [28] Hiriart-Urruty J.-B., Lemaréchal C., Convex Analysis and Minimization Algorithms, I, II, Grundlehren Math. Wiss., 305, 306, Springer, Berlin, 1993 
  29. [29] Jeyakumar V., Li G.Y., Stable zero duality gaps in convex programming: complete dual characterisations with applications to semidefinite programs, J. Math. Anal. Appl., 2009, 360(1), 156–167 http://dx.doi.org/10.1016/j.jmaa.2009.06.043 Zbl1208.90134
  30. [30] Jeyakumar V., Song W., Dinh N., Lee G.M., Stable strong duality in convex optimization, Applied Mathematics Report, 05/22, University of New South Wales, Sydney, 2005 
  31. [31] Kinderlehrer D., Stampacchia G., An Introduction to Variational Inequalities and their Applications, Pure Appl. Math., 88, Academic Press, New York-London, 1980 Zbl0457.35001
  32. [32] Konnov I.V., Schaible S., Duality for equilibrium problems under generalized monotonicity, J. Optim. Theory Appl., 2000, 104(2), 395–408 http://dx.doi.org/10.1023/A:1004665830923 Zbl1016.90066
  33. [33] Mosco U., Dual variational inequalities, J. Math. Anal. Appl., 1972, 40(1), 202–206 http://dx.doi.org/10.1016/0022-247X(72)90043-1 
  34. [34] Rockafellar R.T., Duality and stability in extremum problems involving convex functions, Pacific J. Math., 1967, 21, 167–187 http://dx.doi.org/10.2140/pjm.1967.21.167 Zbl0154.44902
  35. [35] Rockafellar R.T., Convex Analysis, Princeton Math. Ser., 28, Princeton University Press, Princeton, 1970 Zbl0193.18401
  36. [36] Yao J.C., Variational inequalities with generalized monotone operators, Math. Oper. Res., 1994, 19(3), 691–705 http://dx.doi.org/10.1287/moor.19.3.691 Zbl0813.49010
  37. [37] Zălinescu C., Convex Analysis in General Vector Spaces, World Scientific, River Edge, 2002 http://dx.doi.org/10.1142/5021 
  38. [38] Zhang J., Wan C., Xiu N., The dual gap function for variational inequalities, Appl. Math. Optim., 2003, 48(2), 129–148 http://dx.doi.org/10.1007/s00245-003-0771-9 Zbl1048.49007

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.