New Farkas-type constraint qualifications in convex infinite programming
ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 3, p. 580-597

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.

DOI : https://doi.org/10.1051/cocv:2007027
Classification:  90C25,  90C34,  90C46,  90C48
Keywords: convex infinite programming, KKT and saddle point optimality conditions, duality theory, Farkas-type constraint qualification
@article{COCV_2007__13_3_580_0,
author = {Dinh, Nguyen and Goberna, Miguel A. and L\'opez, Marco A. and Son, Ta Quang},
title = {New Farkas-type constraint qualifications in convex infinite programming},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {13},
number = {3},
year = {2007},
pages = {580-597},
doi = {10.1051/cocv:2007027},
zbl = {1126.90059},
mrnumber = {2329178},
language = {en},
url = {http://www.numdam.org/item/COCV_2007__13_3_580_0}
}

Dinh, Nguyen; Goberna, Miguel A.; López, Marco A.; Son, Ta Quang. New Farkas-type constraint qualifications in convex infinite programming. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 3, pp. 580-597. doi : 10.1051/cocv:2007027. http://www.numdam.org/item/COCV_2007__13_3_580_0/

[1] A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer-Verlag, New York (2003). | MR 1931309 | Zbl 1017.49001

[2] J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer-Verlag, New York (2000). | MR 1756264 | Zbl 0966.49001

[3] R.I. Bot and G. Wanka, Farkas-type results with conjugate functions. SIAM J. Optim. 15 (2005) 540-554. | Zbl 1114.90147

[4] R.S. Burachik and V. Jeyakumar, Dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12 (2005) 279-290. | Zbl 1098.49017

[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. | Zbl 0143.42304

[6] F.H. Clarke, A new approach to Lagrange multipliers. Math. Oper. Res. 2 (1976) 165-174. | Zbl 0404.90100

[7] B.D. Craven, Mathematical Programming and Control Theory. Chapman and Hall, London (1978). | MR 515723 | Zbl 0431.90039

[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. | Zbl 1114.90083

[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. | Zbl 1137.90684

[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. | Zbl 0945.90069

[11] M.A. Goberna and M.A. López, Linear Semi-infinite Optimization. J. Wiley, Chichester (1998). | MR 1628195 | Zbl 0909.90257

[12] J. Gwinner, On results of Farkas type. Numer. Funct. Anal. Appl. 9 (1987) 471-520. | Zbl 0598.49017

[13] J.-B. Hiriart Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms I. Springer-Verlag, Berlin (1993). | MR 1261420

[14] V. Jeyakumar, Asymptotic dual conditions characterizing optimality for infinite convex programs. J. Optim. Theory Appl. 93 (1997) 153-165. | Zbl 0901.90158

[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. | Zbl 1038.90061

[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. | Zbl 0856.90128

[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. | Zbl 1046.90059

[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). | MR 467080

[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. | Zbl 1114.90142

[22] W. Li, C. Nahak and I. Singer, Constraint qualification for semi-infinite systems of convex inequalities. SIAM J. Optim. 11 (2000) 31-52. | Zbl 0999.90045

[23] O.L. Mangasarian, Set containment characterization. J. Global Optim. 24 (2002) 473-480. | Zbl 1047.90068

[24] R. Puente and V.N. Vera De Serio, Locally Farkas-Minkowski linear semi-infinite systems. TOP 7 (1999) 103-121. | Zbl 0936.15012

[25] R.T. Rockafellar, Conjugate Duality and Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics 16, SIAM, Philadelphia (1974). | MR 373611 | Zbl 0296.90036

[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. | Zbl 0909.90259

[27] C. Zălinescu, Convex analysis in general vector spaces. World Scientific Publishing Co., NJ (2002). | MR 1921556 | Zbl 1023.46003