The merit function

After a QP subproblem has been solved, new estimates of the NP solution are computed using a linesearch on the augmented Lagrangian merit function

where D is a diagonal matrix of penalty parameters. If (x_k, s_k, p_k) denotes the current solution estimate and (_k, _k, _k) denotes the optimal QP solution, the linesearch determines a step a_k  (0 < a_k 1) such that the new point

gives a sufficient decrease in the merit function.  When necessary, the penalties in D are increased by the minimum-norm perturbation that ensures descent for M [10].  As in NPSOL, sN  is adjusted to minimize the merit function as a function of s prior to the solution of the QP subproblem.  For more details, see [7] and [3].