Abstract
In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.
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This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.)
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Huang, X.X., Yang, X.Q. Further Study on Augmented Lagrangian Duality Theory. J Glob Optim 31, 193–210 (2005). https://doi.org/10.1007/s10898-004-5695-7
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DOI: https://doi.org/10.1007/s10898-004-5695-7