A new superlinearly convergent norm-relaxed method of strongly sub-feasible direction for inequality constrained optimization

Method of feasible directions (MFD) is an important method for solving nonlinearly constrained optimization. However, various types of MFD all need an initial feasible point, which can not be found easily in generally. In addition, the computational cost of some MFD with superlinearly convergent property is rather high. On the other hand, the strongly sub-feasible direction method does not need an initial feasible point, but most of the proposed algorithm do not have the superlinearly convergent property, and can not guarantee that the iteration point is feasible after finite iterations. In this paper, we present a new superlinearly convergent algorithm with arbitrary initial point. At each iteration, a master direction is obtained by solving one direction finding subproblem (DFS), and an auxiliary direction is yielded by an explicit formula. After finite iterations, the iteration point goes into the feasible set and the master direction is a feasible direction of descent. Since a new generalized projection technique is contained in the auxiliary direction formula, under some mild assumptions without the strict complementarity, the global convergence and superlinear convergence of the algorithm can be obtained.