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Liberating the Subgradient Optimality Conditions from Constraint Qualifications

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Abstract

In convex optimization the significance of constraint qualifications is evidenced by the simple duality theory, and the elegant subgradient optimality conditions which completely characterize a minimizer. However, the constraint qualifications do not always hold even for finite dimensional optimization problems and frequently fail for infinite dimensional problems. In the present work we take a broader view of the subgradient optimality conditions by allowing them to depend on a sequence of ε-subgradients at a minimizer and then by letting them to hold in the limit. Liberating the optimality conditions in this way permits us to obtain a complete characterization of optimality without a constraint qualification. As an easy consequence of these results we obtain optimality conditions for conic convex optimization problems without a constraint qualification. We derive these conditions by applying a powerful combination of conjugate analysis and ε-subdifferential calculus. Numerical examples are discussed to illustrate the significance of the sequential conditions.

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Author notes

  1. Z. Y. Wu

    Present address: School of Information Technology Mathematical Sciences, University of Ballarat, Ballarat, Victoria, 3353, Australia

Authors and Affiliations

  1. School of Mathematics, University of New South Wales, Sydney, 2052, Australia

    V. Jeyakumar

  2. Department of Mathematics and Computer Science, Chongqing Normal University, Chongqing, 400047, P. R. China

    Z. Y. Wu

  3. Department of Applied Mathematics, Pukyong National University, Pusan, 608–737, Korea

    G. M. Lee

  4. Department of Mathematics-Informatics, Hochiminh City University of Pedagogy, HCM city, Vietnam

    N. Dinh

Authors
  1. V. Jeyakumar
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  2. Z. Y. Wu
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  3. G. M. Lee
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  4. N. Dinh
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Correspondence to V. Jeyakumar.

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Jeyakumar, V., Wu, Z.Y., Lee, G.M. et al. Liberating the Subgradient Optimality Conditions from Constraint Qualifications. J Glob Optim 36, 127–137 (2006). https://doi.org/10.1007/s10898-006-9003-6

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  • DOI: https://doi.org/10.1007/s10898-006-9003-6

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