Research On A Modified BFGS Method And SQP Method

The BFGS method is a well-known quasi-Newton method for solving unconstained optimization. The method only use the value of objective function and its first-order derivative without computing Hessian matrix, moreover, the method possesses the advantage of fast convengence rate and good numerical results, and so on. Recently, many authors proposed different modified BFGS methods, such as, Fukushima, L. Qi, D. Li, Z. Wei, etc..The sequence quadratic programming method, i.e., SQP method, is a well-known method for solving constrained optimization. This method was proposed by Wilson(1963), but till 1970s it was developed. Han and Powell did one of the most important works for this method. Therefore, SQP method was called Wilson-Han-Powell method. At its iteration, we use B_k to replace W(x_k, λ_k). Recently, this method has been in vogue. Powell [25] proposed BFGS-Newton-SQP method for nonlinearly constrained optimizations. W. Sun [27] gave quasi-Newton-SQP method for general LC~1 constrained optimization problems. And he gave the locally and superlinearly convergent sufficient conditions. But he didn't prove whether the modified BFGS-quasi-Newton-SQP method satisfies the sufficient conditions or not.In this thesis, we propose a new MBFGS algorithm for nonlinearly unconstrained optimizations which based on the study of Z. Wei, et.al [1]. We establish it's global and superlinear convergence without line searches. Enlighten by W. Sun [27], we give a new method (i.e., MSFGS~*-quasi-Newton-SQP method) and establish it's superlinear convergence under suitable conditions.