A class of augmented Lagrangians for equality constraints in nonlinear programming problems

Abstract

In this paper a class of augmented Lagrangians is considered, for solving equality constrained nonlinear optimization problems via unconstrained minimization techniques. This class of augmented Lagrangians is obtained by multiplying the penalty term on the first order necessary optimality condition in a class of augmented Lagrangians of Di Pillo and Grippo by a penalty parameter. Under suitable assumptions, the exactly corresponding relationship is established between the solution of the original constrained problem and the unconstrained minimization of this class of augmented Lagrangians on the product space of problem variables and multipliers for sufficiently large but finite values of penalty parameters. Therefore, a solution of the original constrained problem and the corresponding values of the Lagrange multipliers can be found by performing a single unconstrained minimization of an augmented Lagrangian on the product space of problem variables and multipliers. In particular, for quadratic programming problems with equality constraints, the optimizer is obtained by minimizing a quadratic function on the expanded space.