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Separation Approach for Augmented Lagrangians in Constrained Nonconvex Optimization

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Abstract

This paper aims at showing that the class of augmented Lagrangian functions, introduced by Rockafellar and Wets, can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem; hence, it is part of a more general theory. By means of the image space analysis, local and global saddle-point conditions for the augmented Lagrangian function are investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Under second-order sufficiency conditions in the image space, it is proved that the augmented Lagrangian admits a local saddle point. The existence of a global saddle point is then obtained under additional assumptions that do not require the compactness of the feasible set.

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Authors and Affiliations

  1. Department of Management Science, School of Management, Fudan University, Shanghai, 200433, People’s Republic of China

    H. Z. Luo

  2. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, Zhejiang, 310032, People’s Republic of China

    H. Z. Luo

  3. Dipartimento di Matematica, Universitá di Pisa, Largo B. Pontecorvo 5, 56100, Pisa, Italy

    G. Mastroeni

  4. Department of Mathematics, Hangzou Dianzi University, Hangzhou, Zhejiang, 310018, People’s Republic of China

    H. X. Wu

Authors
  1. H. Z. Luo
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  2. G. Mastroeni
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  3. H. X. Wu
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Corresponding author

Correspondence to G. Mastroeni.

Additional information

Communicated by F. Giannessi.

This work was partially supported by the Natural Science Foundation of Zhejiang Province, Grant Y7080184, and the National Natural Science Foundation of China, Grants 70671064 and 60673177.

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Luo, H.Z., Mastroeni, G. & Wu, H.X. Separation Approach for Augmented Lagrangians in Constrained Nonconvex Optimization. J Optim Theory Appl 144, 275–290 (2010). https://doi.org/10.1007/s10957-009-9598-0

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  • DOI: https://doi.org/10.1007/s10957-009-9598-0

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