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Fenchel Duality and the Strong Conical Hull Intersection Property

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  • F. Deutsch

    (Pennsylvania State University)

  • W. Li

    (Old Dominion University)

  • J. Swetits

    (Old Dominion University)

Abstract

We study a special dual form of a convex minimization problem in a Hilbert space, which is formally suggested by Fenchel dualityand is useful for the Dykstra algorithm. For this special duality problem, we prove that strong duality holds if and only if the collection of underlying constraint sets {C 1,...,C m} has the strong conical hull intersection property. That is, $$\left( {\mathop \cap \limits_1^m C_i - x)^ \circ = \sum\limits_1^m {(C_1 - x} } \right)^ \circ {\text{, for each }}x \in \mathop \cap \limits_1^m C_1$$ where D° denotes the dual cone of D. In general, we can establish weak duality for a convex minimization problem in a Hilbert space by perturbing the constraint sets so that the perturbed sets have the strong conical hull intersection property. This generalizes a result of Gaffke and Mathar.

Suggested Citation

  • F. Deutsch & W. Li & J. Swetits, 1999. "Fenchel Duality and the Strong Conical Hull Intersection Property," Journal of Optimization Theory and Applications, Springer, vol. 102(3), pages 681-695, September.
    • Handle: RePEc:spr:joptap:v:102:y:1999:i:3:d:10.1023_a:1022658308898
    DOI: 10.1023/A:1022658308898
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    References listed on IDEAS

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    1. Norbert Gaffke & Rudolf Mathar, 1989. "A cyclic projection algorithm via duality," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 36(1), pages 29-54, December.
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    Cited by:

    1. Chieu, N.H. & Jeyakumar, V. & Li, G. & Mohebi, H., 2018. "Constraint qualifications for convex optimization without convexity of constraints : New connections and applications to best approximation," European Journal of Operational Research, Elsevier, vol. 265(1), pages 19-25.
    2. H. Mohebi & S. Salkhordeh, 2021. "Robust constrained best approximation with nonconvex constraints," Journal of Global Optimization, Springer, vol. 79(4), pages 885-904, April.

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