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Unformatted text preview: Journal of Convex Analysis Volume 11 (2004), No. 1, 95110 On the Necessity of some Constraint Qualification Conditions in Convex Programming Dan Tiba Weierstrass Institut, Mohrenstr. 39, D10117 Berlin, Germany tiba@wiasberlin.de permanent address: Institute of Mathematics, Romanian Academy, P.O. Box 1764, RO70700, Bucharest, Romania dtiba@imar.ro Constantin Z alinescu University Al. I. Cuza Ia si, Faculty of Mathematics, Bd. Copou Nr. 11, 6600 Ia si, Romania zalinesc@uaic.ro Received February 18, 2002 Revised manuscript received December 11, 2002 In this paper 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 byproduct we obtain a characterization of the metric regularity of a finite family of closed convex sets. Keywords: Convex function, constraint qualification, Lagrange multiplier, metric regularity, normal cone 2000 Mathematics Subject Classification: 49K27, 90C25 1. Introduction Consider the classical convex programming problem minimize g ( x ) s.t. h i ( x ) , i I := { 1 ,...,m } , (P) where g and h i ( i I ) are convex functions defined on the normed vector space X . We are interested by the weakest hypotheses that ensure the characterization of a solution x of (P) by KarushKuhnTucker conditions, i.e. , the existence of 1 ,..., m 0, called Lagrange multipliers, such that x is a solution of the (unconstrained) minimization problem minimize g ( x ) + X m i =1 i h i ( x ) s.t. x X (UP) and i h i ( x ) = 0 for every i I . There are several known assumptions of this type in the literature, called constraint qualification (CQ) conditions. The mostly used seems to be Slaters CQ: e x, i I : h i ( e x ) < . (SCQ) Denoting by C the set { x X  h i ( x ) i I } of admissible solutions of (P) and by I ( x ) the set { i I  h i ( x ) = 0 } of active constraints at x C , and assuming the functions ISSN 09446532 / $ 2.50 c Heldermann Verlag 96 D. Tiba, C. Zalinescu / On the Necessity of some Constraint Qualification ... h i to be finitevalued and differentiable, other conditions are: { h i ( x )  i I ( x ) } is linearly independent, (LICQ) the MangasarianFromovitz CQ e u, i I ( x ) : h i ( x )( e u ) < , (MFCQ) or Abadies CQ cone( C x ) = { u  h i ( x )( u ) i I ( x ) } . (ACQ) As noted by Li [19], (ACQ) is equivalent to the condition N ( C, x ) = X i I ( x ) i h i ( x ) i i I ( x ) , (ACQ ) where N ( C, x ) is the normal cone of C at x . Abadies CQ is, consequently, a particular case of the basic constraint qualification introduced by HiriartUrruty and Lemar echal [15] in the case of nondifferentiable convex minimization problems:...
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