An augmented Lagrangian affine scaling method for nonlinear programming

Abstract

In this paper, we propose an augmented Lagrangian affine scaling (ALAS) algorithm for general nonlinear programming, for which a quadratic approximation to the augmented Lagrangian is minimized at each iteration. Different from the classical sequential quadratic programming (SQP), the linearization of nonlinear constraints is put into the penalty term of this quadratic approximation, which results in smooth objective of the subproblem and avoids possible inconsistency among the linearized constraints and trust region constraint. By applying affine scaling techniques to handle the strict bound constraints, an active set type affine scaling trust region subproblem is proposed. Through special strategies of updating Lagrange multipliers and adjusting penalty parameters, this subproblem is able to reduce the objective function value and feasibility errors in an adaptive well-balanced way. Global convergence of the ALAS algorithm is established under mild assumptions. Furthermore, boundedness of the penalty parameter is analyzed under certain conditions. Preliminary numerical experiments of ALAS are performed for solving a set of general constrained nonlinear optimization problems in CUTEr problem library.