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A modified barrier-augmented Lagrangian method for constrained minimization. (English) Zbl 0951.90042
Summary: We present and analyze an interior-exterior augmented Lagrangian method for solving constrained optimization problems with both inequality and equality constraints. This method, the Modified Barrier-Augmented Lagrangian (MBAL) method, is a combination of the modified barrier and the augmented Lagrangian methods. It is based on the MBAL function, which treats inequality constraints with a modified barrier term and equalities with an augmented Lagrangian term. The MBAL method alternatively minimizes the MBAL function in the primal space and updates the Lagrange multipliers. For a large enough fixed barrier-penalty parameter the MBAL method is shown to converge $$Q$$-linearly under the standard second-order optimality conditions. $$Q$$-superlinear convergence can be achieved by increasing the barrier-penalty parameter after each Lagrange multiplier update. We consider a dual problem that is based on the MBAL function. We prove a basic duality theorem for it and show that it has several important properties that fail to hold for the dual based on the classical Lagrangian.

##### MSC:
 90C30 Nonlinear programming 90C46 Optimality conditions and duality in mathematical programming
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