Strongly Sub-Feasible Sequential Systems Of Linear Equations Algorithm With Bidirectional Line Search For Constrained Optimization

This thesis discusses the sequential systems of linear equations (SSLE) algorithms for solving nonlinear inequality constrained optimization prob-lems. Firstly, by improving the algorithm of [Li J L, Jian J B. OR TRANSACTIONS,2003,7(2)], a strongly sub-feasible sequence of lin-ear equations algorithm is proposed. Each iteration, the search direction is generated by a simple structure systems of linear equations, or is re-placed by an auxiliary direction which can ensure strong convergence and be canceled automatically after a finite number of iterations. By traditional Armijo line search to get step size, the number of constraints which are satisfied by iteration points is nondecreasing, and the value of objective function does not increase too much. Under some suitable con-ditions, the iteration point can get into the feasible region after a finite number of iterations. Secondly, this thesis proposes a new line search technique which is named as bidirectional line search. The main idea is to get possibly large step size by line search from forward (the direction which step size decreases gradually) or backward (the direction which step size increases gradually). So the step size is not limited to less then or equal to one, and it can be greater than one. This technology greatly enhanced the speed of convergence, especially for large-scale nonlinear optimization problem. With bidirectional search technique, a strongly sub-feasible linear equations algorithm is presented. Under some certain conditions, the algorithm has global convergence and superlinear conver-gence as well as quadratic convergence. Finally, we do some numerical experiments, which show that the algorithm and the bidirectional line search are effective, especially for large-scale problems.