On The Superlinear Convergence Of A Penalty-free Method

Penalty-type methods and penalty-free methods for solving constrained optimization problem all may be – at least if no second order correction steps are used – affected by the well known Maratos effect. By the Maratos effect a full SQP-step can lead to an increase of both the objective function value and the measure of constraint violation even arbitrarily close to a regular minimizer. This will prohibit fast local convergence.The usual methods to avoid Maratos effect are second order correction method and nonmonotone technique. The computation of second order correction steps can be cumbersome and the nonmonotone framework will complicate the algorithm implementation in our experience. It is very important value in theory and practice to study using Lagrangian function directly, neither to adopt second order correction method nor to use nonmonotone technique, to avoid Maratos effect.In this paper, we propose a kind of penalty-free algorithm with trust region framework for nonlinear equality constrained optimization problem. At each iteration, the trial step is composed of a normal step and a tangent step. According to the relations among the predicted reduction on Lagrangian function, the measure of constraint violation and trust region radius, we determine whether the current iteration is f-type iteration or c-type iteration not. For f-type iteration, the algorithm requires that the value of Lagrangian function is sufficient descent. For c-type iteration, it requires only that the measure of constraint violation is sufficient descent. The algorithm does not need the feasibility restoration phase and does not use second order correction method and nonmonotone technique. Under the usual assumptions, we analyzed the well definedness of the algorithm and proved the global convergence of the algorithm. Under the second order sufficient condition, the algorithm is one-step superlinearly convergent. Finally, a preliminary numerical results are reported for equality constrained optimization problems in CUTEr collection. We also use the performance profile to display the performance based on the computational results between Lancelot with default settings and the algorithm presented in this paper.