jca0373 - Journal of Convex Analysis Volume 11 (2004), No....

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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, D-10117 Berlin, Germany tiba@wias-berlin.de permanent address: Institute of Mathematics, Romanian Academy, P.O. Box 1-764, 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 by-product 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 Karush-Kuhn-Tucker 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 0944-6532 / $ 2.50 c Heldermann Verlag 96 D. Tiba, C. Zalinescu / On the Necessity of some Constraint Qualification ... h i to be finite-valued and differentiable, other conditions are: { h i ( x ) | i I ( x ) } is linearly independent, (LICQ) the Mangasarian-Fromovitz 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 Hiriart-Urruty and Lemar echal [15] in the case of nondifferentiable convex minimization problems:...
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jca0373 - Journal of Convex Analysis Volume 11 (2004), No....

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