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Conical open mapping theorems and regularity

Bauschke, Heinz H. and Borwein, Jonathan M. (1999) Conical open mapping theorems and regularity. In: Proceedings of the Centre for Mathematics and its Applications (Australian National University) , March 1998, National Symposium on Functional Analysis, Optimization and Applications.

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      Abstract

      Suppose $T$ is a continuous linear operator between two Hilbert spaces $X$ and $Y$ and let $K$ be a closed convex nonempty cone in $X$. We investigate the possible existence of $\delta > 0$ such that $\delta B_Y \cap T(K) \subseteq T(B_X \cap K)$, where $B_X,B_Y$ denote the closed unit balls in $X$ and $Y$ respectively. This property, which we call openness relative to $K$, is a generalization of the classical openness of linear operators. We relate relative openness to Jameson's property~(G), to the strong conical hull intersection property, to bounded linear regularity, and to metric regularity. Our results allow a simple construction of two closed convex cones that have the strong conical hull intersection property but fail to be boundedly linearly regular.

      Item Type: Conference or Workshop Item (Paper)
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: bounded linear regularity, linear regularity, conical hull intersection property, metric regularity, normal cone, open mapping theorem, property (G), strong CHIP, strong conical hull intersection property
      Subjects: 90-xx Economics, operations research, programming, games > 90Cxx Mathematical programming
      47-xx Operator theory > 47Bxx Special classes of linear operators
      49-xx Calculus of variations and optimal control; optimization > 49Nxx Miscellaneous topics
      52-xx Convex and discrete geometry > 52Axx General convexity
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 25 Nov 2003
      Last Modified: 01 Mar 2015 17:29
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/208

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