Characterizations of ɛ-duality gap statements for constrained optimization problems

Horaţiu-Vasile Boncea; Sorin-Mihai Grad

Open Mathematics (2013)

  • Volume: 11, Issue: 11, page 2020-2033
  • ISSN: 2391-5455

Abstract

top
In this paper we present different regularity conditions that equivalently characterize various ɛ-duality gap statements (with ɛ ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ɛ-subdifferentials. When ɛ = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature.

How to cite

top

Horaţiu-Vasile Boncea, and Sorin-Mihai Grad. "Characterizations of ɛ-duality gap statements for constrained optimization problems." Open Mathematics 11.11 (2013): 2020-2033. <http://eudml.org/doc/269358>.

@article{Horaţiu2013,
abstract = {In this paper we present different regularity conditions that equivalently characterize various ɛ-duality gap statements (with ɛ ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ɛ-subdifferentials. When ɛ = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature.},
author = {Horaţiu-Vasile Boncea, Sorin-Mihai Grad},
journal = {Open Mathematics},
keywords = {Conjugate functions; ɛ-duality gap; Constraint qualifications; Lagrange dual problems; Fenchel-Lagrange dual problems; conjugate functions; -duality gap; constraint qualifications; epigraphs; -sudifferentials},
language = {eng},
number = {11},
pages = {2020-2033},
title = {Characterizations of ɛ-duality gap statements for constrained optimization problems},
url = {http://eudml.org/doc/269358},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Horaţiu-Vasile Boncea
AU - Sorin-Mihai Grad
TI - Characterizations of ɛ-duality gap statements for constrained optimization problems
JO - Open Mathematics
PY - 2013
VL - 11
IS - 11
SP - 2020
EP - 2033
AB - In this paper we present different regularity conditions that equivalently characterize various ɛ-duality gap statements (with ɛ ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ɛ-subdifferentials. When ɛ = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature.
LA - eng
KW - Conjugate functions; ɛ-duality gap; Constraint qualifications; Lagrange dual problems; Fenchel-Lagrange dual problems; conjugate functions; -duality gap; constraint qualifications; epigraphs; -sudifferentials
UR - http://eudml.org/doc/269358
ER -

References

top
  1. [1] Anbo Y., Nonstandard arguments and the characterization of independence in generic structures, RIMS Kôkyûroku, 2009, 1646, 4–17 
  2. [2] 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
  3. [3] Boţ R.I., Grad S.-M., Wanka G., Maximal monotonicity for the precomposition with a linear operator, SIAM J. Optim., 2006, 17(4), 1239–1252 Zbl1133.47038
  4. [4] Boţ R.I., Grad S.-M., Wanka G., Weaker constraint qualifications in maximal monotonicity, Numer. Funct. Anal. Optim., 2007, 28(1–2), 27–41 Zbl1119.47051
  5. [5] Boţ R.I., Grad S.-M., Wanka G., A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces, Math. Nachr., 2008, 281(8), 1088–1107 http://dx.doi.org/10.1002/mana.200510662 Zbl1155.49019
  6. [6] Boţ R.I., Grad S.-M., Wanka G., New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces, Nonlinear Anal., 2008, 69(1), 323–336 http://dx.doi.org/10.1016/j.na.2007.05.021 Zbl1142.49015
  7. [7] Boţ R.I., Grad S.-M., Wanka G., On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl., 2008, 337(2), 1315–1325 http://dx.doi.org/10.1016/j.jmaa.2007.04.071 Zbl1160.90004
  8. [8] Boţ R.I., Grad S.-M., Wanka G., Duality in Vector Optimization, Vector Optim., Springer, Berlin, 2009 Zbl1177.90355
  9. [9] Boţ R.I., Grad S.-M., Wanka G., Generalized Moreau-Rockafellar results for composed convex functions, Optimization, 2009, 58(7), 917–933 http://dx.doi.org/10.1080/02331930902945082 Zbl1201.90154
  10. [10] Boţ R.I., Wanka G., Farkas-type results with conjugate functions, SIAM J. Optim., 2005, 15(2), 540–554 http://dx.doi.org/10.1137/030602332 Zbl1114.90147
  11. [11] Boţ R.I., Wanka G., An alternative formulation for a new closed cone constraint qualification, Nonlinear Anal., 2006, 64(6), 1367–1381 http://dx.doi.org/10.1016/j.na.2005.06.041 Zbl1105.46052
  12. [12] 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
  13. [13] Fang D.H., Li C., Ng K.F., Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming, Nonlinear Anal., 2010, 73(5), 1143–1159 http://dx.doi.org/10.1016/j.na.2010.04.020 Zbl1218.90200
  14. [14] Friedman H.M., A way out, In: One Hundred Years of Russell’s Paradox, de Gruyter Ser. Log. Appl., 6, de Gruyter, Berlin, 2004, 49–84 
  15. [15] Jeyakumar V., Li G.Y., New dual constraint qualifications characterizing zero duality gaps of convex programs and semidefinite programs, Nonlinear Anal., 2009, 71(12), e2239–e2249 http://dx.doi.org/10.1016/j.na.2009.05.009 Zbl1239.90084
  16. [16] 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
  17. [17] Li C., Fang D., López G., López M.A., Stable and total Fenchel duality for convex optimization problems in locally convex spaces, SIAM J. Optim., 2009, 20(2), 1032–1051 http://dx.doi.org/10.1137/080734352 Zbl1189.49051
  18. [18] Rubinov A.M., Glover B.M., Quasiconvexity via two step functions, In: Generalized Convexity, Generalized Monotonicity: Recent Results, Luminy, June 17–21, 1996, Nonconvex Optim. Appl., 27, Kluwer, Dordrecht, 1998, 159–183 http://dx.doi.org/10.1007/978-1-4613-3341-8_5 

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.