We introduce a twice differentiable augmented Lagrangian for nonlinear optimization with general inequality constraints and show that a strict local minimizer of the original problem is an approximate strict local solution of the augmented Lagrangian. A novel augmented Lagrangian method of multipliers (ALM) is then presented. Our method is originated from a generalization of the Hestenes-Powell augmented Lagrangian, and is a combination of the augmented Lagrangian and the interior-point technique. It shares a similar algorithmic framework with existing ALMs for optimization with inequality constraints, but it can use the second derivatives and does not depend on projections on the set of inequality constraints. In each iteration, our method solves a twice continuously differentiable unconstrained optimization subproblem on primal variables. The dual iterates, penalty and smoothing parameters are updated adaptively. The global and local convergence are analyzed. Without assuming any constraint qualification, it is proved that the proposed method has strong global convergence. The method may converge to either a Karush-Kuhn-Tucker (KKT) point or a singular stationary point when the converging point is a minimizer. It may also converge to an infeasible stationary point of nonlinear program when the problem is infeasible. Furthermore, our method is capable of rapidly detecting the possible infeasibility of the solved problem. Under suitable conditions, it is locally linearly convergent to the KKT point, which is consistent with ALMs for optimization with equality constraints. The preliminary numerical experiments on some small benchmark test problems demonstrate our theoretical results. 

　　Publication: 

　　Mathematics of Computation, 92 (2023), 1301-1330 

　　DOI: 10.1007/s00220-022-04609-1 

　　Author: 

　　Xin-Wei Liu 

　　Institute of Mathematics, Hebei University of Technology, Tianjin 300401, People’s Republic of China 

　　Yu-Hong Dai 

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China 

　　Email address: dyh@lsec.cc.ac.cn 

　　Ya-Kui Huang 

　　Institute of Mathematics, Hebei University of Technology, Tianjin 300401, People’s Republic of China 

　　Jie Sun 

　　Institute of Mathematics, Hebei University of Technology, Tianjin 300401, People’s Republic of China; and School of Business, National University of Singapore, Singapore 119245, Singapore