In CCoM Report 22-01, Gill and Zhang propose a primal-dual path-following method for general nonlinearly constrained optimization that combines a shifted primal-dual path-following method with a projected-search method for bound-constrained optimization. The method involves the computation of an approximate Newton direction for a primal-dual penalty-barrier function that incorporates shifts on both the primal and dual variables. This note concerns the formulation of approximate Newton equations for a nonlinear optimization problem in general form. These equations may be used in conjunction with a projected-search method to force convergence from an arbitrary starting point. It is shown that under certain conditions, the approximate Newton equations are equivalent to a regularized form of the conventional primal-dual path-following equations.
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