【作者】 裴杰；

【导师】 刘陶文；

【作者基本信息】 湖南大学 ， 运筹学与控制论， 2008， 硕士

【摘要】 本文研究用既约Hessian SQP方法求解等式约束问题.与一般SQP方法相比,既约Hessian SQP方法能节省大量的存储空间.因此,这类方法能有效地求解较大规模的等式约束问题.然而已有的这类方法的全局收敛性分析需请求较强的条件,如假定Lagrange函数的既约Hessian矩阵序列的一致正定性,而这种假定通常很难被满足.因此,在没有上述假定的情况下研究用既约Hessian方法求解约束问题具有重要的理论与实际意义.本文提出了既约Hessian SQP方法的两种修正.在第一章中,我们介绍了非线性规划的基本理论,包括BFGS校正技术,然后给出了既约Hessian SQP方法的基本结构.在第二章我们首先推广了求解无约束问题的MBFGS校正技术,并将其应用到求解等式约束优化问题中,提出了一个修正的既约Hessian SQP方法,并且在较弱的条件下建立了全局收敛性结果.分析表明该方法同时具有局部R-线性收敛性和2-步超线性收敛速度.我们在第三章研究了结合MBFGS与CBFGS两种方法的修正既约Hessian SQP方法,提出带混合校正技术的既约Hessian SQP方法,这种方法与第二章的方法具有相同的收敛性及优点.在第四章我们针对第二章,第三章所提出的算法进行了数值实验,数值结果表明本文所提出的算法是有效的.数值实验结果的比较表明第三章中的算法在各个方面均比第二章中的算法要更为有效.更多还原

【Abstract】 In this thesis, we are concerned with equality constrained optimization prob-lems. It is well-known that the sequential quadratic programming(SQP) methodsare welcome for small and middle size problem. For large scale problem, how-ever, SQP method may not e?cient as even not over all. The reduced HessianSQP method can be applied to solve large scale problem due to its requirementof less space for storage. Existing reduced Hessian SQP methods are practicallye?cient. However, the conditions for a reduced Hessian SQP method are rigorous.In general the uniform positive definition of the reduced Hessian approximate isnecessary.In this thesis, basing on an MBFGS update for unconstrained optimization,we propose a way to construct quasi-Newton update for the reduced Hessian ap-proximating. In Chapter One, we introduce some well known iteration methodsincluding BFGS method. We then introduce the basis steps of the reduced HessianSQP method. In Chapter Two, we first introduce the MBFGS formula and thenapply it to the equality constrained problems. Under mild conditions, we prove theglobal convergence of the related reduced Hessian SQP method. We also obtainthe R-linear and two step super linear convergence of the proposed method. InChapter Three, by combining the MBFGS and CBFGS update formula, we pro-pose another update formula for the reduced Hessian approximation. The relatemethod retains the same convergence property as the method in Chapter Two.In Chapter Four, we also do some numerical experiments, the results show thee?ciency of the proposed methods.更多还原