New Farkas-type constraint qualifications in convex infinite programming

Nguyen Dinh; Miguel A. Goberna; Marco A. López; Ta Quang Son

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 3, page 580-597
  • ISSN: 1292-8119

Abstract

top
This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given closed convex set which satisfy a (possibly infinite) convex system. Moreover, all the involved functions are assumed to be convex, lower semicontinuous and proper (but not necessarily real-valued). The key result in the paper is the characterization of those reverse-convex inequalities which are consequence of the constraints system. As a byproduct of this new versions of Farkas' lemma we also characterize the containment of convex sets in reverse-convex sets. The main results in the paper are obtained under a suitable Farkas-type constraint qualifications and/or a certain closedness assumption.

How to cite

top

Dinh, Nguyen, et al. "New Farkas-type constraint qualifications in convex infinite programming." ESAIM: Control, Optimisation and Calculus of Variations 13.3 (2007): 580-597. <http://eudml.org/doc/249989>.

@article{Dinh2007,
abstract = { This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given closed convex set which satisfy a (possibly infinite) convex system. Moreover, all the involved functions are assumed to be convex, lower semicontinuous and proper (but not necessarily real-valued). The key result in the paper is the characterization of those reverse-convex inequalities which are consequence of the constraints system. As a byproduct of this new versions of Farkas' lemma we also characterize the containment of convex sets in reverse-convex sets. The main results in the paper are obtained under a suitable Farkas-type constraint qualifications and/or a certain closedness assumption. },
author = {Dinh, Nguyen, Goberna, Miguel A., López, Marco A., Son, Ta Quang},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Convex infinite programming; KKT and saddle point optimality conditions; duality theory; Farkas-type constraint qualification; convex infinite programming},
language = {eng},
month = {6},
number = {3},
pages = {580-597},
publisher = {EDP Sciences},
title = {New Farkas-type constraint qualifications in convex infinite programming},
url = {http://eudml.org/doc/249989},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Dinh, Nguyen
AU - Goberna, Miguel A.
AU - López, Marco A.
AU - Son, Ta Quang
TI - New Farkas-type constraint qualifications in convex infinite programming
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/6//
PB - EDP Sciences
VL - 13
IS - 3
SP - 580
EP - 597
AB - This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given closed convex set which satisfy a (possibly infinite) convex system. Moreover, all the involved functions are assumed to be convex, lower semicontinuous and proper (but not necessarily real-valued). The key result in the paper is the characterization of those reverse-convex inequalities which are consequence of the constraints system. As a byproduct of this new versions of Farkas' lemma we also characterize the containment of convex sets in reverse-convex sets. The main results in the paper are obtained under a suitable Farkas-type constraint qualifications and/or a certain closedness assumption.
LA - eng
KW - Convex infinite programming; KKT and saddle point optimality conditions; duality theory; Farkas-type constraint qualification; convex infinite programming
UR - http://eudml.org/doc/249989
ER -

References

top
  1. A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer-Verlag, New York (2003).  
  2. J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer-Verlag, New York (2000).  
  3. R.I. Bot and G. Wanka, Farkas-type results with conjugate functions. SIAM J. Optim.15 (2005) 540–554.  
  4. R.S. Burachik and V. Jeyakumar, Dual condition for the convex subdifferential sum formula with applications. J. Convex Anal.12 (2005) 279–290.  
  5. A. Charnes, W.W. Cooper and K.O. Kortanek, On representations of semi-infinite programs which have no duality gaps. Manage. Sci.12 (1965) 113–121.  
  6. F.H. Clarke, A new approach to Lagrange multipliers. Math. Oper. Res. 2 (1976) 165–174.  
  7. B.D. Craven, Mathematical Programming and Control Theory. Chapman and Hall, London (1978).  
  8. N. Dinh, V. Jeyakumar and G.M. Lee, Sequential Lagrangian conditions for convex programs with applications to semidefinite programming. J. Optim. Theory Appl.125 (2005) 85–112.  
  9. N. Dinh, M.A. Goberna and M.A. López, From linear to convex systems: consistency, Farkas' lemma and applications. J. Convex Anal.13 (2006) 279–290.  
  10. M.D. Fajardo and M.A. López, Locally Farkas-Minkowski systems in convex semi-infinite programming. J. Optim. Theory Appl.103 (1999) 313–335.  
  11. M.A. Goberna and M.A. López, Linear Semi-infinite Optimization. J. Wiley, Chichester (1998).  
  12. J. Gwinner, On results of Farkas type. Numer. Funct. Anal. Appl.9 (1987) 471–520.  
  13. J.-B. Hiriart Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms I. Springer-Verlag, Berlin (1993).  
  14. V. Jeyakumar, Asymptotic dual conditions characterizing optimality for infinite convex programs. J. Optim. Theory Appl.93 (1997) 153–165.  
  15. V. Jeyakumar, Farkas' lemma: Generalizations, in Encyclopedia of Optimization II, C.A. Floudas and P. Pardalos Eds., Kluwer, Dordrecht (2001) 87–91.  
  16. V. Jeyakumar, Characterizing set containments involving infinite convex constraints and reverse-convex constraints. SIAM J. Optim.13 (2003) 947–959.  
  17. V. Jeyakumar, A.M. Rubinov, B.M. Glover and Y. Ishizuka, Inequality systems and global optimization. J. Math. Anal. Appl.202 (1996) 900–919.  
  18. V. Jeyakumar, G.M. Lee and N. Dinh, New sequential Lagrange multiplier conditions characterizing optimality without constraint qualifications for convex programs. SIAM J. Optim.14 (2003) 534–547.  
  19. V. Jeyakumar, N. Dinh and G.M. Lee, A new closed cone constraint qualification for convex optimization, Applied Mathematics Research Report AMR04/8, UNSW, 2004. Unpublished manuscript. http://www.maths.unsw.edu.au/applied/reports/amr08.html  
  20. P.-J. Laurent, Approximation et optimization. Hermann, Paris (1972).  
  21. C. Li and K.F. Ng, On constraint qualification for an infinite system of convex inequalities in a Banach space. SIAM J. Optim.15 (2005) 488–512.  
  22. W. Li, C. Nahak and I. Singer, Constraint qualification for semi-infinite systems of convex inequalities. SIAM J. Optim.11 (2000) 31–52.  
  23. O.L. Mangasarian, Set containment characterization. J. Global Optim.24 (2002) 473–480.  
  24. R. Puente and V.N. Vera de Serio, Locally Farkas-Minkowski linear semi-infinite systems. TOP7 (1999) 103–121.  
  25. R.T. Rockafellar, Conjugate Duality and Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics16, SIAM, Philadelphia (1974).  
  26. A. Shapiro, First and second order optimality conditions and perturbation analysis of semi-infinite programming problems, in Semi-Infinite Programming, R. Reemtsen and J. Rückmann Eds., Kluwer, Dordrecht (1998) 103–133.  
  27. C. Zălinescu, Convex analysis in general vector spaces. World Scientific Publishing Co., NJ (2002).  

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.