The strong conical hull intersection property for convex programming Article

First Online: 07 June 2005 Received: 05 September 2004 Accepted: 14 April 2005 DOI :
10.1007/s10107-005-0605-4

Cite this article as: Jeyakumar, V. Math. Program. (2006) 106: 81. doi:10.1007/s10107-005-0605-4
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

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, is a continuous convex function and g :X →Y 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 The author is grateful to the referees for their constructive comments and valuable suggestions which have contributed to the final preparation of the paper.

References 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)

MATH MathSciNet CrossRef Google Scholar 2.

Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part I: quasi–relative interiors and duality. Math. Progr.

57 , 15–48 (1992)

MATH CrossRef Google Scholar 3.

Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge, 2004

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)

MathSciNet CrossRef Google Scholar 5.

Burachik, R.S., Jeyakumar, V.: A new geometric condition for Fenchel's duality in infinite dimensions. Math. Progr. Series B (to appear)

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 1998

7.

Deutsch, F.: Best approximation in inner product spaces. Springer-Verlag, New York, 2001

8.

Deutsch, F., Li, W., Swetits, J.: Fenchel duality and the strong conical hull intersection property. J. Optim. Theory Appl.

102 , 681–695 (1999)

MATH MathSciNet CrossRef Google Scholar 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)

MATH MathSciNet CrossRef Google Scholar 10.

Henrion, R., Jourani, A.: Subdifferential conditions for calmness of convex constraints. SIAM J. Optim.

13 , 520–534 (2002)

MATH MathSciNet CrossRef Google Scholar 11.

Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim.

13 (2), 603–618 (2002)

MathSciNet CrossRef Google Scholar 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. 43–92

13.

Hiriart-Urruty, J.B., Lemarechal, C.: Convex Analysis and Minimization Algorithms. Springer-Verlag, Berlin, 1993

14.

Hiriart-Urruty, J.B., Phelps, R.R.: Subdifferential calculus using ∈-subdifferentials. J. Funct. Anal.

18 , 154–166 (1993)

MathSciNet CrossRef Google Scholar 15.

Jeyakumar, V., Mohebi, H.: A global approach to nonlinearly constrained best approximation. Numer. Funct. Anal. and Optim. (to appear)

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)

MathSciNet CrossRef Google Scholar 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, 2005

18.

Jeyakumar, V., Rubinov, A.M., Glover, B.M., Ishizuka, Y.: Inequality systems and global optimization. J. Math. Anal. Appl.

202 , 900–919 (1996)

MATH MathSciNet CrossRef Google Scholar 19.

Jeyakumar, V., Wolkowicz, H.: Generalizations of Slater's constraint qualification for infinite convex programs. Math. Progr.

57 (1), 85–102 (1992)

MathSciNet CrossRef Google Scholar 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)

CrossRef Google Scholar 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)

MATH MathSciNet CrossRef Google Scholar 22.

Ng, K.F., Song, W.: Fenchel duality in infinite-dimensional setting and its applications. Nonlinear Anal.

25 , 845–858 (2003)

CrossRef Google Scholar 23.

Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of semidefinite programming. Int Series Oper Res Management Sci 27 , Kluwer Academic Publishers, Dordrecht, 2000

24.

Strömberg, T.: The operation of infimal convolution. Diss. Math.

352 , 1–61 (1996)

Google Scholar 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)

26.

Zalinescu, C.: Convex analysis in general vector spaces. World Scientific, London, 2002

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Authors and Affiliations 1. Department of Applied Mathematics University of New South Wales Sydney Australia