Results 1  10
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32
Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization
, 1997
"... The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental notions occur in various branches of convex optimization (constrained approximation, convex feasibility pr ..."
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Cited by 31 (3 self)
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The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental notions occur in various branches of convex optimization (constrained approximation, convex feasibility problems, linear inequalities, for instance). It is shown that the standard constraint qualification from convex analysis implies bounded linear regularity, which in turn yields the strong conical hull intersection property. Jameson's duality for two cones, which relates bounded linear regularity to property (G), is rederived and refined. For polyhedral cones, a statement dual to Hoffman's error bound result is obtained. A sharpening of a result on error bounds for convex inequalities by Auslender and Crouzeix is presented. Finally, for two subspaces, property (G) is quantified by the angle between the subspaces.
On error bounds for lower semicontinuous functions on Banach Spaces
 MATH. PROGRAMMING, SER. A
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SUFFICIENT CONDITIONS FOR ERROR BOUNDS
, 2001
"... For a lower semicontinuous (l.s.c.) inequality system on a Banach space, it is shown that error bounds hold, provided every element in an abstract subdifferential of the constraint function at each point outside the solution set is norm bounded away from zero. A sufficient condition for a global e ..."
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Cited by 18 (6 self)
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For a lower semicontinuous (l.s.c.) inequality system on a Banach space, it is shown that error bounds hold, provided every element in an abstract subdifferential of the constraint function at each point outside the solution set is norm bounded away from zero. A sufficient condition for a global error bound to exist is also given for an l.s.c. inequality system on a real normed linear space. It turns out that a global error bound closely relates to metric regularity, which is useful for presenting sufficient conditions for an l.s.c. system to be regular at sets. Under the generalized Slater condition, a continuous convex system on R n is proved to be metrically regular at bounded sets.
Subdifferential conditions for calmness of convex constraints
 SIAM J. Optim
"... Abstract. We study subdifferential conditions of the calmness property for multifunctions representing convex constraint systems in a Banach space. Extending earlier work in finite dimensions [R. Henrion and J. Outrata, J. Math. Anal. Appl., 258 (2001), pp. 110–130], we show that, in contrast to the ..."
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Cited by 16 (1 self)
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Abstract. We study subdifferential conditions of the calmness property for multifunctions representing convex constraint systems in a Banach space. Extending earlier work in finite dimensions [R. Henrion and J. Outrata, J. Math. Anal. Appl., 258 (2001), pp. 110–130], we show that, in contrast to the stronger Aubin property of a multifunction (or metric regularity of its inverse), calmness can be ensured by corresponding weaker constraint qualifications, which are based only on boundaries of subdifferentials and normal cones rather than on the full objects. Most of the results can be immediately interpreted in the context of error bounds.
METRIC SUBREGULARITY AND CONSTRAINT QUALIFICATIONS FOR CONVEX GENERALIZED EQUATIONS IN BANACH SPACES
"... Several notions of constraint qualifications are generalized from the setting of convex inequality systems to that of convex generalized equations. This is done and investigated in terms of the coderivatives and the normal cones, and thereby we provide some characterizations for convex generalized e ..."
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Cited by 15 (4 self)
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Several notions of constraint qualifications are generalized from the setting of convex inequality systems to that of convex generalized equations. This is done and investigated in terms of the coderivatives and the normal cones, and thereby we provide some characterizations for convex generalized equations to have the metric subregularity. As applications, we establish formulas of the modulus of calmness and provide several characterizations of the calmness. Extending the classical concept of extreme boundary, we introduce a notion of recession cores of closed convex sets. Using this concept, we establish global metric subregularity (i.e. error bound) results for generalized equations.
The SECQ, Linear Regularity and the Strong CHIP for Infinite System of Closed Convex Sets in Normed Linear Spaces
"... We consider a (finite or infinite) family of closed convex sets with nonempty intersection in a normed space. A property relating their epigraphs with their intersection's epigraph is studied, and its relations to other constraint qualifications (such as the linear regularity, the strong CHIP a ..."
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Cited by 14 (5 self)
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We consider a (finite or infinite) family of closed convex sets with nonempty intersection in a normed space. A property relating their epigraphs with their intersection's epigraph is studied, and its relations to other constraint qualifications (such as the linear regularity, the strong CHIP and Jameson's (G)property) are established. With suitable continuity assumption we show how this property can be ensured from the corresponding property of some of its finite subfamilies.
Regularities and Their Relations to Error Bounds
"... In this paper, we mainly study various notions of regularity for a finite collection {C1, · · · , Cm} of closed convex subsets of a Banach space X and their relations with other fundamental concepts. We show that a proper lower semicontinuous function f on X has a Lipschitz error bound (resp., Υ ..."
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Cited by 11 (1 self)
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In this paper, we mainly study various notions of regularity for a finite collection {C1, · · · , Cm} of closed convex subsets of a Banach space X and their relations with other fundamental concepts. We show that a proper lower semicontinuous function f on X has a Lipschitz error bound (resp., Υerror bound) if and only if the pair {epi(f), X ×{0}} of sets in the product space X × R is linearly regular (resp., regular). Similar results for multifunctions are also established. Next, we prove that {C1, · · · , Cm} is linearly regular if and only if it has the strong CHIP and the collection {NC1(z), · · · , NCm(z)} of normal cones at z has property (G) for each z ∈ C: = ∩ m i=1Ci. Provided that C1 is a closed convex cone and that C2 = Y is a closed vector subspace of X, we show that {C1, Y} is linearly regular if and only if there exists α> 0 such that each positive (relative to the order induced by C1) linear functional on Y of norm one can be extended to a positive linear functional on X with norm bounded by α. Similar characterization is given in terms of normal cones.
LINEAR REGULARITY FOR A COLLECTION OF SUBSMOOTH SETS IN BANACH SPACES
"... Using variational analysis, we study the linear regularity for a collection of finitely many closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of finitely many closed convex sets to the possibly nonconvex setting. Moreover the sharpest linear r ..."
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Cited by 11 (5 self)
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Using variational analysis, we study the linear regularity for a collection of finitely many closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of finitely many closed convex sets to the possibly nonconvex setting. Moreover the sharpest linear regularity constant can also be dually represented under the subsmoothness assumption.
Firstorder and secondorder conditions for error bounds
 SIAM J. Optim
"... Abstract. For a lower semicontinuous function f on a Banach space X, we studythe existence of a positive scalar µ such that the distance function dS associated with the solution set S of f(x) ≤ 0 satisfies dS(x) ≤ µ max{f(x), 0} for each point x in a neighborhood of some point x0 in X with f(x) &l ..."
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Cited by 10 (5 self)
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Abstract. For a lower semicontinuous function f on a Banach space X, we studythe existence of a positive scalar µ such that the distance function dS associated with the solution set S of f(x) ≤ 0 satisfies dS(x) ≤ µ max{f(x), 0} for each point x in a neighborhood of some point x0 in X with f(x) <ɛfor some 0 <ɛ ≤ +∞. We give several sufficient conditions for this in terms of an abstract subdifferential and the Dini derivatives of f. In a Hilbert space we further present some secondorder conditions. We also establish the corresponding results for a system of inequalities, equalities, and an abstract constraint set.
METRIC SUBREGULARITY AND CALMNESS FOR NONCONVEX GENERALIZED EQUATIONS IN BANACH SPACES
, 2010
"... This paper concerns a generalized equation defined by a closed multifunction between Banach spaces, and we employ variational analysis techniques to provide sufficient and/or necessary conditions for a generalized equation to have the metric subregularity (i.e., local error bounds for the concerne ..."
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Cited by 10 (1 self)
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This paper concerns a generalized equation defined by a closed multifunction between Banach spaces, and we employ variational analysis techniques to provide sufficient and/or necessary conditions for a generalized equation to have the metric subregularity (i.e., local error bounds for the concerned multifunction) in general Banach spaces. Following the approach of Ioffe [Trans. Amer. Math. Soc., 251 (1979), pp. 61–69] who studied the numerical function case, our conditions are described in terms of coderivatives of the concerned multifunction at points outside the solution set. Motivated by the existing modulus representation and pointbased criteria for the metric regularity, we establish the corresponding results for the metric subregularity. In the Asplund space case, sharper results are obtained.