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Mathematical Programming

, Volume 86, Issue 1, pp 135–160 | Cite as

Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization

  • Heinz H. Bauschke
  • Jonathan M. Borwein
  • Wu Li

Abstract.

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 re-derived 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.

Key words: angle – asymptotic constraint qualification – basic constraint qualification – bounded linear regularity – CHIP – conical hull intersection property – convex feasibility problem – convex inequalities – constrained best approximation – error bound – Friedrichs angle – Hoffman’s error bound – linear inequalities – linear regularity – orthogonal projection – property (G) Mathematics Subject Classification (1991): Primary 90C25; Secondary 15A39, 41A29, 46A40, 46C05, 90C31 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Heinz H. Bauschke
    • 1
  • Jonathan M. Borwein
    • 2
  • Wu Li
    • 3
  1. 1.Department of Mathematics and Statistics, Okanagan University College, Kelowna, British Columbia V1V 1V7, Canada, e-mail: bauschke@cecm.sfu.ca.CA
  2. 2.Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada, e-mail: jborwein@cecm.sfu.ca.CA
  3. 3.Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia 23529, USA, e-mail: wuli@math.odu.eduUS

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