A Feasible Active Set QP-Free Method for Nonlinear Programming

We propose a monotone descent active set QP-free method for inequality constrained optimization that ensures the feasibility of all iterates and allows for iterates on the boundary of the feasible set. The study is motivated by the Facchinei--Fischer--Kanzow active set identification technique for nonlinear programming and variational inequalities [F. Facchinei, A. Fischer, and C. Kanzow, SIAM J. Optim., 9 (1999), pp. 14-32]. Distinguishing features of the proposed method compared with existing QP-free methods include lower subproblem costs and a fast convergence rate under milder assumptions. Specifically, four reduced linear systems with a common coefficient matrix involving only constraints in a working set are solved at each iteration. To determine the working set, the method makes use of multipliers from the last iteration, eliminating the need to compute a new estimate, and no additional linear systems are solved to select linearly independent constraint gradients. A new technique is presented to avoid possible ill-conditioned Newton systems caused by dual degeneracy. It is shown that the method converges globally to KKT points under the linear independence constraint qualification (LICQ), and the asymptotic rate of convergence is Q-superlinear under an additional strong second-order sufficient condition (SSOSC) without strict complementarity.