Abstract

We realize a study of various constraint qualification conditions for the existence of Lagrange multipliers for convex minimization problems in general normed vector spaces; it is based on a new formula for the normal cone to the constraint set, on local metric regularity and a metric regularity property on bounded subsets. As a by-product we obtain a characterization of the metric regularity of a finite family of closed convex sets.

Author information

Contact details are reproduced from the original publication and may be historical.

Dan Tiba

Weierstrass Institut, Mohrenstr. 39, D-10117 Berlin, Germany
and: Institute of Mathematics, Romanian Academy of Sciences, P.O. Box 1-764, 70700 Bucharest, Romania

dtiba@imar.ro

Constantin Zălinescu

University "Al. I. Cuza", Faculty of Mathematics, Bd. Copou Nr. 11, 6600 Iaşi, Romania

zalinesc@uaic.ro

Keywords

  • convex function
  • constraint qualification
  • Lagrange multiplier
  • metric regularity
  • normal cone

Mathematics Subject Classification

49K27, 90C25

References

  1. V. Barbu: Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden (1976).https://doi.org/10.1007/978-94-010-1537-0Google Scholar
  2. V. Barbu, Th. Precupanu: Convexity and Optimization in Banach Spaces, Editura Academiei (Bucureşti) and Sijthoff & Noordhoff Int. Publ. (Alphen aan de Rijn) (1978).Google Scholar
  3. V. Barbu, Th. Precupanu: Convexity and Optimization in Banach Spaces, Editura Academiei (Bucureşti) and D. Reidel Publishing Company (Dordrecht) (1986).Google Scholar
  4. H. H. Bauschke, J. M. Borwein: On the convergence of von Neumann’s alternating projection algorithm for two sets, Set-Valued Anal. 1 (1993) 185–212.https://doi.org/10.1007/BF01027691Google Scholar
  5. H. H. Bauschke, J. M. Borwein: On projection algorithms for solving convex feasibility problems, SIAM Rev. 38 (1996) 367–426.https://doi.org/10.1137/S0036144593251710Google Scholar
  6. H. H. Bauschke, J. M. Borwein, W. Li: Strong conical hull intersection property, bounded linear regularity and Jameson’s property (G), and error bounds in convex optimization, Math. Programming 86 (1999) 135–160.https://doi.org/10.1007/s101070050083Google Scholar
  7. H. H. Bauschke, J. M. Borwein, P. Tseng: Bounded linear regularity, strong CHIP, and CHIP are distinct properties, J. Convex Analysis 7(2) (2000) 395–412.https://doi.org/10.68381/jca07020Google Scholar
  8. F. Bonnans, A. Shapiro: Perturbation Analysis of Optimization Problems, Springer, New York (2000).https://doi.org/10.1007/978-1-4612-1394-9Google Scholar
  9. J. M. Borwein, J. D. Vanderwerff: Convex functions of Legendre type in general Banach spaces, J. Convex Analysis 8(2) (2001) 569–581.https://doi.org/10.68381/jca08030Google Scholar
  10. O. Cornejo, A. Jourani, C. Zălinescu: Conditioning and upper-Lipschitz inverse subdifferentials in nonsmooth optimization problems, J. Optim. Theory Appl. 95 (1997) 127–148.https://doi.org/10.1023/A:1022687412779Google Scholar
  11. S. Deng: Global error bounds for convex inequality systems in Banach spaces, SIAM J. Control Optim. 36 (1998) 1240–1249.https://doi.org/10.1137/S0363012995293645Google Scholar
  12. J. Diestel: Geometry of Banach Spaces, Lecture Notes in Math. 485, Springer-Verlag, Berlin (1975).https://doi.org/10.1007/BFb0082079Google Scholar
  13. I. Ekeland, R. Temam: Analyse Convexe et Problèmes Variationnels, Collection Études Mathématiques, Dunod (1974).Google Scholar
  14. J. R. Giles: Convex Analysis with Application in the Differentiation of Convex Functions, Research Notes in Mathematics 58, Pitman Advanced Publishing Program, Boston, Mass. (1982).Google Scholar
  15. J.-B. Hiriart-Urruty, Cl. Lemarechal: Convex Analysis and Minimization Algorithms, Springer Verlag, Berlin (1993).https://doi.org/10.1007/978-3-662-02796-7Google Scholar
  16. D. Klatte, W. Li: Asymptotic constraint qualifications and global error bounds for convex inequalities, Math. Programming 84 (1999) 137–160.https://doi.org/10.1007/s10107980002aGoogle Scholar
  17. P. J. Laurent: Approximation et Optimisation, Hermann, Paris (1972).Google Scholar
  18. B. Lemaire: Bounded diagonally stationary sequences in convex optimization, J. Convex Analysis 1(1) (1994) 75–86.https://doi.org/10.68381/jca01005Google Scholar
  19. W. Li: Abadie’s constraint qualification, metric regularity and error bounds for differentiable convex inequalities, SIAM J. Control Optimiz. 7 (1998) 966–978.https://doi.org/10.1137/S1052623495287927Google Scholar
  20. A. S. Lewis, J.-S. Pang: Error bounds for convex inequality systems, Proc. 5th Int. Symp. on Generalized Convexity held in Luminy, June 17–21, 1996, J.-P. Crouzeix, J.-E. Martinez-Legaz, M. Volle (eds.), Kluwer Academic Publishers (Dordrecht) (1998) 75–110.https://doi.org/10.1007/978-1-4613-3341-8_3Google Scholar
  21. J.-S. Pang: Error bounds in mathematical programming, Math. Programming 79 (1997) 299–332.https://doi.org/10.1016/S0025-5610(97)00042-7Google Scholar
  22. J.-P. Penot, C. Zălinescu: Harmonic sum and duality, J. Convex Analysis 7(1) (2000) 95–113.https://doi.org/10.68381/jca07005Google Scholar
  23. B. N. Pshenichnyi: Convex Analysis and Extremal Problems (Russian), Nauka, Moskow (1980).Google Scholar
  24. S. M. Robinson: An application of error bounds for convex programming in a linear space, SIAM J. Control 13 (1975) 271–273.https://doi.org/10.1137/0313015Google Scholar
  25. R. T. Rockafellar: Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, Princeton, New Jersey (1970)Google Scholar
  26. D. Tiba: On the convex programming, Preprint Nr. 2, Institute of Mathematics, Romanian Academy, Bucharest (1995).Google Scholar
  27. D. Tiba: Sur l’existence des multiplicateurs de Lagrange, Preprint Nr. 2, Institute of Mathematics, Romanian Academy, Bucharest (1996).Google Scholar
  28. D. Tiba, M. Bergounioux: General optimality conditions for constrained convex control problems, SIAM J. Control Optimiz. 34(2) (1996) 698–711.https://doi.org/10.1137/S0363012994261987Google Scholar
  29. V. M. Tikhomirov: Convex analysis, in: Analysis II, R. V. Gamkrelidze (ed.), Encyclopaedia of Mathematical Sciences 14, Springer Verlag, Berlin (1990) 1–92.https://doi.org/10.1007/978-3-642-61267-1_1Google Scholar
  30. C. Zălinescu: Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces, Proc. 12th Baikal Int. Conf. on Optimization Methods and their Applications, Irkutsk, Russia, 2001, Irkutsk Univ. Press 272–284.Google Scholar
  31. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

D. Tiba, C. Zălinescu. “On the Necessity of some Constraint Qualification Conditions in Convex Programming.” Journal of Convex Analysis 11 (2004), No. 1, 95–110. https://doi.org/10.68381/jca11007

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Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.