Abstract
In this paper, by using an augmented Lagrangian approach, we obtain several sufficient conditions for the existence of augmented Lagrange multipliers of a cone constrained optimization problem in Banach spaces, where the corresponding augmenting function is assumed to have a valley at zero. Furthermore, we deal with the relationship of saddle points, augmented Lagrange multipliers, and zero duality gap property between the cone constrained optimization problem and its augmented Lagrangian dual problem.
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The authors would like to thank the referees for the valuable suggestions which have improved the early version of the manuscript.
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Yu Ying Zhou: Research of this author was supported by the Natural Science Foundation of China (11071180, 11171247).
Jin Chuan Zhou: Research of this author was supported by Natural Science Foundation of China (11101248, 11271233), Shandong Province Natural Science Foundation (ZR2010AQ026), and Young Teacher Support Program of Shandong University of Technology.
Xiao Qi Yang: Research of this author was supported by Natural Science Foundation of China (10831009) and grants from the Research Grants Council of Hong Kong (PolyU 5306/11E).
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Zhou, Y.Y., Zhou, J.C. & Yang, X.Q. Existence of augmented Lagrange multipliers for cone constrained optimization problems. J Glob Optim 58, 243–260 (2014). https://doi.org/10.1007/s10898-013-0046-1
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DOI: https://doi.org/10.1007/s10898-013-0046-1