A Penalty-free Line Search Without Maratos Effect

The method for solving nonlinear constrained optimization problems is divided into two categories:one is penalty-type methods,the main feature of which is the use of penalty factors;one is penalty-free methods.They all may be affected by the 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 when the iteration is arbitrarily close to a regular minimizer.This will prohibit fast local convergence.Second order correction method and nonmonotone technique are two usual methods to avoid Maratos effect.However,it is necessary to study the method using neither second order correct method nor nonmonotone technique to avoid Maratos effect.In this paper,we propose a new penalty-free algorithm with linear search for nonlinear equality constrained optimization problem.Firstly,we settle a linear pro-gramming problem,whose solution can improve the measure of constraint violation.Secondly,we solve a quadratic programming subproblem which is always compatible and whose solution will improve the measure of the optimality.We choose some con-vex combination between those two solutions as a line search direction.We will use the Lagrangian function instead of the objective function to avoid the Maratos effec-t.According to the relations among the predicted reduction on Lagrangian function,the measure of constraint violation and step size,we determine whether the current iteration is 1-type iteration or c-type iteration not.For 1-type iteration,the algorithm requires that the value of Lagrangian function is sufficient descent.For c-type itera-tion,it requires that the measure of constraint violation is sufficient descent.Under the suitable assumptions,the method is globally convergent and it is one-step superlineaxly convergent.Finally,some preliminary numerical results are reported.