Abstract
We introduce some approximate optimal solutions and a generalized augmented Lagrangian in nonlinear programming, establish dual function and dual problem based on the generalized augmented Lagrangian, obtain approximate KKT necessary optimality condition of the generalized augmented Lagrangian dual problem, prove that the approximate stationary points of generalized augmented Lagrangian problem converge to that of the original problem. Our results improve and generalize some known results.
Highlights
It is well known that dual method and penalty function method are popular methods in solving nonlinear optimization problems
Many constrained optimization problems can be formulated as an unconstrained optimization problem by dual method or penalty function method
In this paper, based on the results in [4, 10, 14], we investigate the possibility of obtaining the various versions of approximate solutions to a constrained optimization problem by solving an unconstrained programming problem formulated by using a generalized augmented Lagrangian function
Summary
It is well known that dual method and penalty function method are popular methods in solving nonlinear optimization problems. Based on the augmented Lagrangian, a strong duality result without any convexity requirement in the primal problem was obtained under mild conditions. Chen et al [2] and Huang and Yang [3] used augmented Lagrangian functions to construct the set-valued dual functions and corresponding dual problems and obtained weak and strong duality results of multiobjective optimization problem. In this paper, based on the results in [4, 10, 14], we investigate the possibility of obtaining the various versions of approximate solutions to a constrained optimization problem by solving an unconstrained programming problem formulated by using a generalized augmented Lagrangian function. An approximate KKT optimality condition is obtained for a kind of approximate solutions to the generalized augmented Lagrangian problem.
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