【作者】 刘利英；

【导师】 韦增欣；

【作者基本信息】 广西大学 ， 应用数学， 2006， 硕士

【摘要】 BFGS方法是求解无约束优化问题的著名的拟牛顿方法。它只需利用目标函数值和一阶导数的信息，而不需要计算Hessian矩阵，且具有收敛速度快和数值结果好等优点。近年来，许多学者给出了不同的修改的BFGS方法，如Fukushima，祁力群，李董辉，韦增欣等。 著名的序列二次规划方法，简称SQP方法，是求解非线性约束优化问题的一类非常重要的方法。该方法最早由Wilson(1963)提出，但直到20世纪70年代中期才引起人们的重视并得到发展．其中，韩世平和Powell的工作非常重要，因此，又称SQP方法为Wilson-Han-Powell方法。它在每次迭代中用一修正的矩阵B_k代替W(x_k，λ_k)。近年来，该方法有较好的发展趋势，Powell[25]提出BFGS-Newton-SQP方法用来求解非线性约束优化问题。孙文瑜[27]给出了求解半光滑约束优化问题的quasi-Newton-SQP型方法及其全局收敛和超线性收敛的充分必要条件，但是并没有证明修改的BFGS-quasi-Newton-SQP型方法是否满足条件。 本文是在韦增欣等[1]研究的基础上，给出一个新的MBFGS算法，并证明该算法在无线搜索条件下的全局收敛性和超线性收敛性。另外，受到孙文瑜的启发，本文构造出求解约束优化问题的一种新方法(MBFGS~*-quasi-Newton-SQP方法)，并在适当的条件下证明此算法的超线性收敛性。更多还原

【Abstract】 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.更多还原