Mathematical Programming

, Volume 106, Issue 1, pp 81–92 | Cite as

The strong conical hull intersection property for convex programming

Article

Abstract

The strong conical hull intersection property (CHIP) is a geometric property of a collection of finitely many closed convex intersecting sets. This basic property, which was introduced by Deutsch et al. in 1997, is one of the central ingredients in the study of constrained interpolation and best approximation. In this paper we establish that the strong CHIP of intersecting sets of constraints is the key characterizing property for optimality and strong duality of convex programming problems. We first show that a sharpened strong CHIP is necessary and sufficient for a complete Lagrange multiplier characterization of optimality for the convex programming model problem Open image in new window

where C is a closed convex subset of a Banach space X, S is a closed convex cone which does not necessarily have non-empty interior, Y is a Banach space, Open image in new window is a continuous convex function and g:XY is a continuous S-convex function. We also show that the strong CHIP completely characterizes the strong duality for partially finite convex programs, where Y is finite dimensional and g(x)=−Ax+b and S is a polyhedral convex cone. Global sufficient conditions which are strictly weaker than the Slater type conditions are given for the strong CHIP and for the sharpened strong CHIP.

Keywords

Strong conical hull intersection property global constraint qualification strong duality optimality conditions constrained approximation 

Mathematics Subject Classification

41A65 41A29 90C30 

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References

  1. 1.
    Bauschke, H.H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization. Math. Progr. 86, 135–160 (1999)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part I: quasi–relative interiors and duality. Math. Progr. 57, 15–48 (1992)MATHCrossRefGoogle Scholar
  3. 3.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge, 2004Google Scholar
  4. 4.
    Burachik, R.S., Jeyakumar, V.: A simple closure condition for the normal cone intersection formula. Proc. Amer. Math. Soc. 133 (6), 1741–1748 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Burachik, R.S., Jeyakumar, V.: A new geometric condition for Fenchel's duality in infinite dimensions. Math. Progr. Series B (to appear)Google Scholar
  6. 6.
    Deutsch, F.: The role of conical hull intersection property in convex optimization and approximation. In: Approximation Theory IX, Chui C.K., Schumaker L.L. (eds.), Vanderbilt University Press, Nashville, TN 1998Google Scholar
  7. 7.
    Deutsch, F.: Best approximation in inner product spaces. Springer-Verlag, New York, 2001Google Scholar
  8. 8.
    Deutsch, F., Li, W., Swetits, J.: Fenchel duality and the strong conical hull intersection property. J. Optim. Theory Appl. 102, 681–695 (1999)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Deutsch, F., Li, W., Ward, J.D.: Best approximation from the intersection of a closed convex set and a polyhedron in Hilbert space, weak Slater conditions, and the strong conical hull intersection property. SIAM J. Optim. 10, 252–268 (1999)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Henrion, R., Jourani, A.: Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13, 520–534 (2002)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13 (2), 603–618 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hiriart-Urruty, J.B.: ∈-subdifferential calculus. In: Convex Analysis and Optimization, Aubin J.P., Vinter R.B. (eds.), Research Notes in Mathematics 57, Pitman, 1982, pp. 43–92Google Scholar
  13. 13.
    Hiriart-Urruty, J.B., Lemarechal, C.: Convex Analysis and Minimization Algorithms. Springer-Verlag, Berlin, 1993Google Scholar
  14. 14.
    Hiriart-Urruty, J.B., Phelps, R.R.: Subdifferential calculus using ∈-subdifferentials. J. Funct. Anal. 18, 154–166 (1993)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jeyakumar, V., Mohebi, H.: A global approach to nonlinearly constrained best approximation. Numer. Funct. Anal. and Optim. (to appear)Google Scholar
  16. 16.
    Jeyakumar, V., Lee, G.M., Dinh, N.: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J. Optim. 14 (2), 534–547 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jeyakumar, V., Song, W., Dinh, N., Lee, G.M.: Stable strong duality in convex optimization. Applied Mathematics Preprint, University of New South Wales, Sydney, 2005Google Scholar
  18. 18.
    Jeyakumar, V., Rubinov, A.M., Glover, B.M., Ishizuka, Y.: Inequality systems and global optimization. J. Math. Anal. Appl. 202, 900–919 (1996)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Jeyakumar, V., Wolkowicz, H.: Generalizations of Slater's constraint qualification for infinite convex programs. Math. Progr. 57 (1), 85–102 (1992)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li, C., Jin, X.: Nonlinearly constrained best approximation in Hilbert spaces: the strong CHIP, and the basic constraint qualification. SIAM J. Optim. 13 (1), 228–239 (2002)CrossRefGoogle Scholar
  21. 21.
    Li, C., Ng, K.F.: Constraint qualification, the strong CHIP and best approximation with convex constraints in Banach spaces. SIAM J. Optim. 14, 584–607 (2003)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Ng, K.F., Song, W.: Fenchel duality in infinite-dimensional setting and its applications. Nonlinear Anal. 25, 845–858 (2003)CrossRefGoogle Scholar
  23. 23.
    Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of semidefinite programming. Int Series Oper Res Management Sci 27, Kluwer Academic Publishers, Dordrecht, 2000Google Scholar
  24. 24.
    Strömberg, T.: The operation of infimal convolution. Diss. Math. 352, 1–61 (1996)Google Scholar
  25. 25.
    Tiba, D., Zalinescu, C.: On the necessity of some constraint qualification condition in convex programming. J. Convex Anal. 11 (1 & 2), 95–110 (2004)Google Scholar
  26. 26.
    Zalinescu, C.: Convex analysis in general vector spaces. World Scientific, London, 2002Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of New South WalesSydneyAustralia

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