# 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

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topDinh, 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- A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer-Verlag, New York (2003). Zbl1017.49001
- J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer-Verlag, New York (2000). Zbl0966.49001
- R.I. Bot and G. Wanka, Farkas-type results with conjugate functions. SIAM J. Optim.15 (2005) 540–554. Zbl1114.90147
- R.S. Burachik and V. Jeyakumar, Dual condition for the convex subdifferential sum formula with applications. J. Convex Anal.12 (2005) 279–290. Zbl1098.49017
- 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. Zbl0143.42304
- F.H. Clarke, A new approach to Lagrange multipliers. Math. Oper. Res. 2 (1976) 165–174. Zbl0404.90100
- B.D. Craven, Mathematical Programming and Control Theory. Chapman and Hall, London (1978). Zbl0431.90039
- 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. Zbl1114.90083
- 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. Zbl1137.90684
- M.D. Fajardo and M.A. López, Locally Farkas-Minkowski systems in convex semi-infinite programming. J. Optim. Theory Appl.103 (1999) 313–335. Zbl0945.90069
- M.A. Goberna and M.A. López, Linear Semi-infinite Optimization. J. Wiley, Chichester (1998). Zbl0909.90257
- J. Gwinner, On results of Farkas type. Numer. Funct. Anal. Appl.9 (1987) 471–520. Zbl0598.49017
- J.-B. Hiriart Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms I. Springer-Verlag, Berlin (1993).
- V. Jeyakumar, Asymptotic dual conditions characterizing optimality for infinite convex programs. J. Optim. Theory Appl.93 (1997) 153–165. Zbl0901.90158
- V. Jeyakumar, Farkas' lemma: Generalizations, in Encyclopedia of Optimization II, C.A. Floudas and P. Pardalos Eds., Kluwer, Dordrecht (2001) 87–91.
- V. Jeyakumar, Characterizing set containments involving infinite convex constraints and reverse-convex constraints. SIAM J. Optim.13 (2003) 947–959. Zbl1038.90061
- V. Jeyakumar, A.M. Rubinov, B.M. Glover and Y. Ishizuka, Inequality systems and global optimization. J. Math. Anal. Appl.202 (1996) 900–919. Zbl0856.90128
- 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. Zbl1046.90059
- 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
- P.-J. Laurent, Approximation et optimization. Hermann, Paris (1972).
- 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. Zbl1114.90142
- W. Li, C. Nahak and I. Singer, Constraint qualification for semi-infinite systems of convex inequalities. SIAM J. Optim.11 (2000) 31–52.
- O.L. Mangasarian, Set containment characterization. J. Global Optim.24 (2002) 473–480. Zbl1047.90068
- R. Puente and V.N. Vera de Serio, Locally Farkas-Minkowski linear semi-infinite systems. TOP7 (1999) 103–121.
- R.T. Rockafellar, Conjugate Duality and Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics16, SIAM, Philadelphia (1974).
- 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. Zbl0909.90259
- C. Zălinescu, Convex analysis in general vector spaces. World Scientific Publishing Co., NJ (2002). Zbl1023.46003

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