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We introduce a twice differentiable augmented Lagrangian for nonlinearoptimization with general inequality constraints and show that a strict localminimizer of the original problem is an approximate strict local solution ofthe augmented Lagrangian. A novel augmented Lagrangian method of multipliers(ALM) is then presented. Our method is originated from a generalization of theHetenes-Powell augmented Lagrangian, and is a combination of the augmentedLagrangian and the interior-point technique. It shares a similar algorithmicframework with existing ALMs for optimization with inequality constraints, butit can use the second derivatives and does not depend on projections on the setof inequality constraints. In each iteration, our method solves a twicecontinuously differentiable unconstrained optimization subproblem on primalvariables. The dual iterates, penalty and smoothing parameters are updatedadaptively. The global and local convergence are analyzed. Without assuming anyconstraint qualification, it is proved that the proposed method has strongglobal convergence. The method may converge to either a Kurash-Kuhn-Tucker(KKT) point or a singular stationary point when the converging point is aminimizer. It may also converge to an infeasible stationary point of nonlinearprogram when the problem is infeasible. Furthermore, our method is capable ofrapidly detecting the possible infeasibility of the solved problem. Undersuitable conditions, it is locally linearly convergent to the KKT point, whichis consistent with ALMs for optimization with equality constraints. Thepreliminary numerical experiments on some small benchmark test problemsdemonstrate our theoretical results.

A novel augmented Lagrangian method of multipliers for optimization with general inequality constraints