Study Of Several Algorithms For Nonlinear Optimization Problems

The paper is organized as follows: In the first chapter, we briefly review the developing history of SQP method and SSLE method, and introduce some new achievements in this field in recent years. Then the global and locally superlinear convergent conditions are considered and the problems existed in these types of algorithms are presented. At the same time, we also discuss some solutions of these problems. Moreover, we mainly introduce a filter method, which is most widely used in SQP-type method and give some results and new achievements in recent years.In the second chapter, by combining the filter method and active-set identification technique, a new filter-SSLE algorithm for inequality constrained optimization problems is proposed. The algorithm needs to solve only two or three systems of linear equations with the same coefficient matrix at each iteration to get the search direction and it is proved to be globally and superlinearly convergent under mild conditions.In the thirld chapter, we propose a new infeasible SSLE algorithm for NLP. The new algorithm is based on filter method and an extended active set identification technique. At each iteration, the most two linear equations with the same coefficient matrix need to be solved and when the iterates are close to the solution only one of them is involved. Moreover, the equations involve only constraints in the working set and those not in the working set are totally neglected, which reduces the problem size greatly. In the new algorithm, we incorporate the filter method which, compared with the penalty function method, avoids the difficulty in choosing the penalty parameter for different problems and makes the new algorithm more applicable. In particular, it has both global convergence and local superlinear convergence with less computation and weaker assumptions.