An Efficient Method For Solving Constrained Nonlinear Programming Problems

A new efficient method for solving nonlinear programming (NLP) problem is studied in this paper. This is a hybrid method of multiplier method and quasi-Newton method. In order to avoid solving QP subproblem, we develop a new nonlinear system which is equivalent to KKT conditions of the problem. NCP function is used in the nonlinear system such that the nonnegativity of some variables is avoided. In order to guarantee the global convergence, a method of multiplier is inserted in the iterative process. When the iterative point is not near to the optimal point, we use the approximating negative gradient obtained by the finite-difference technique as the descent direction such that it can guarantee the quick descent quality. When the iterative point is near to the optimal point, we develop a linear system by Newton-equation. We use BFGS updating formula to approach the Hessian matrix. The global convergence of the hybrid algorithm is proved and some numerical tests for the algorithm are given. The theoretical and numerical results show that the hybrid method is efficient.