Abstract. We introduce a notion of quasi-regularity for points with respect to the inclusion F (x) ∈ C where F is a nonlinear Frechét differentiable function from R v to R m . When C is the set of minimum points of a convex real-valued function h on R m and F satisfies the L-average Lipschitz condition of Wang, we use the majorizing function technique to establish the semi-local linear/quadratic convergence of sequences generated by the Gauss-Newton method (with quasi-regular initial points) for the convex composite function h • F . Results are new even when the initial point is regular and F is Lipschitz.
Abstract. For an inequality system defined by a possibly infinite family of proper functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we obtain characterizations of those reverse-convex inequalities which are consequence of the constrained system, and we provide necessary and/or sufficient conditions for a stable Farkas lemma to hold. Similarly, we provide characterizations for constrained minimization problems to have the strong or strong stable Lagrangian dualities. Several known results in the conic programming problem are extended and improved.
Abstract. We consider a ( nite or in nite) family of closed convex sets with nonempty i n tersection in a normed space. A property relating their epigraphs with their intersection's epigraph is studied, and its relations to other constraint quali cations (such as the linear regularity, the strong CHIP and Jameson's (G)-property) are established. With suitable continuity assumption we s h o w h o w t h i s p r o p e r t y can be ensured from the corresponding property of some of its nite subfamilies.
Abstract. For an inequality system defined by an infinite family of proper convex functions, we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions and study relationships between these new constraint qualifications and other well known constraint qualifications including the basic constraint qualification studied by Hiriart-Urrutty and Lemarechal, Li, Nahak and Singer. Extensions of known results to more general settings are presented, and applications to particular important problems, such as conic programming and approximation theory, are also studied.
Abstract. We establish several set-valued function versions of Ekeland's variational principle and hence provide some sufficient conditions ensuring the existence of error bounds for inequality systems defined by finitely many lower semicontinuous functions.
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