A Superlinearly Convergent Norm-Relaxed Method Of Strongly Sub-Feasible Directions For Nonlinearly Constrained Optimization

The method of feasible directions (MFD for short) is an important method for solving nonlin-early constrained optimization. Due to some important advantages such as the descent property, feasibility of all iterations, computational efficiency and so on, the method of feasible directions is especially popular in engineering design optimization problems. In the recent years, some new algorithms of MFD such as the generalized Norm-Relaxed method of feasible directions have been proposed and investigated widely, and these algorithms possess good convergence rate. However, various algorithms of MFD have a common shortcoming, that is, the initial iteration point must be feasible, so an auxiliary procedure must be considered for finding an initial feasible point, i.e., a system of nonlinear inequalities must be solved before using the method of feasible directions, which is not easy to be finished in generally, especially for large scale problems. In addition, the su-perlinearly convergent property discussed in the proposed algorithms of MFD were obtained under the strict complementarity assumption, which is rather strong and difficult for testing.In this thesis, we make a deep investigation on the method of feasible directions. Combining the generalized norm-relaxed method of feasible directions with the idea of strongly sub-feasible direction method, we present a new convergent algorithm with arbitrary initial point for inequality constrained optimization. At each iteration, a master direction is obtained by solving one direction finding subproblem which always possesses a solution, and an auxiliary direction is yielded by an explicit formula. After finite iterations, the iteration point gets 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. In the end, the numerical experiment results show that the proposed algorithm is effective.