Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we consider the formulation of subproblems in which the objective function is a generalization of the Hestenes-Powell augmented Lagrangian function. The main feature of the generalized function is that it is minimized with respect to both the primal and the dual variables simultaneously. The benefits of this approach include: (i) the ability to control the quality of the dual variables during the solution of the subproblem; (ii) the availability of improved dual estimates on early termination of the subproblem; and (iii) the ability to regularize the subproblem by imposing explicit bounds on the dual variables. We propose two primal-dual variants of conventional primal methods: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual L1-linearly constrained Lagrangian (pdL1LCL) method. Finally, a new sequential quadratic programming (pdSQP) method is proposed that uses the primal-dual augmented Lagrangian as a merit function. Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we discuss the formulation of subproblems in which the objective is a primal-dual generalization of the Hestenes-Powell augmented Lagrangian function. This generalization has the crucial feature that it is minimized with respect to both the primal and the dual variables simultaneously. A benefit of this approach is that the quality of the dual variables is monitored explicitly during the solution of the subproblem. Moreover, each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of conventional primal methods are proposed: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual l_1 linearly constrained Lagrangian method.
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