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New Farkas-type constraintqualifications in convex infinite programming

Published online by Cambridge University Press:  20 June 2007

Nguyen Dinh ,
Miguel A. Goberna ,
Marco A. López  and
Ta Quang Son
Nguyen Dinh
Affiliation:
Department of Mathematics, International University, Vietnam National University-HCM city, Linh Trung ward, Thu Duc district, Ho Chi Minh city, Vietnam.
Miguel A. Goberna
Affiliation:
Department of Statistics and Operations Research, University of Alicante, 03080 Alicante, Spain; Marco.Antonio@ua.es
Marco A. López
Affiliation:
Department of Statistics and Operations Research, University of Alicante, 03080 Alicante, Spain; Marco.Antonio@ua.es
Ta Quang Son
Affiliation:
Nha Trang College of Education, Nha Trang, Vietnam.
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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.

Keywords

Convex infinite programmingKKT and saddle point optimality conditionsduality theoryFarkas-type constraint qualification
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 3 , July 2007 , pp. 580 - 597
Copyright
© EDP Sciences, SMAI, 2007

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References

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