【作者】 张杰；

【导师】 张立卫；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2007， 硕士

【摘要】 经典的Lagrange函数(即关于乘子向量与约束映射均是线性的函数)在凸规划对偶理论的研究中起重要的作用，尤其线性规划与二次规划的对偶理论要通过经典的Lagrange函数来表达。但对于非凸规划而言，基于经典Larange函数的对偶问题与原始问题存在对偶间隙，因此研究经典Lagrange函数的各种变形就成为人们关注的热点。非线性Lagrange函数是经典Lagrange函数的修正形式，它关于乘子向量或约束函数是非线性函数，基于非线性Lagrange函数建立的求解优化问题的对偶方法即为非线性Lagrange方法。由于对偶算法对原始变量的可行性没有限制，因此非线性Lagrange方法在求解约束优化问题中扮演着重要的角色。基于非线性Lagrange方法的成熟和非凸半定规划在实际问题中的广泛应用，我们尝试把这种方法推广到求解非凸半定规划中。我们构造了基于一种非线性函数的对偶算法来解决非凸半定规划问题，并证明了其局部收敛性，即由非线性Lagrange算法产生的序列局部收敛到原问题的KKT解，并建立了参数解的误差估计式，本文取得的主要结果可概括如下：1．本文第二章归纳和总结了非凸半定规划的最优性条件，这些条件是下一章构造的对偶算法的理论分析所必备的。这章首先介绍了抽象约束优化问题的最优性条件，之后把它用到非凸半定规划中去。2．本文第三章给出了非凸半定规划的一个非线性Lagrange算法，并证明了它的收敛性。即证明了在适当条件下，罚参数存在一个阀值，当罚参数小于这一阀值时，由非线性Lagrange算法产生的序列局部收敛到原问题的Karush-Kuhn-Thcker(KKT)解，并建立了参数解的误差估计式。更多还原

【Abstract】 Classical Lagrangians in which the multiplier vectors and the constraint mappings areinvloved in linear ways, play an important role in studies on the duality theories of con-vex programmings,especially that of linear programmings and quadratic programmingswhich should be express through Classical Lagrangians. But in nonconvex programmings,the primal problems and the duality problems which are based on Classical Lagrangianshave duality gaps. So many sholars are become more and more interested in studyingon the varies of Classical Larangians.Nonlinear Lagrangians are variants of the classicalLagrangian, in which the multiplier vectors or constraint functions are involved in nonlin-ear ways. Nonlinear Lagrange methods are dual methods based on nonlinear Lagrangiansfor solving optimization problems.As dual methods usually require no restrictions on thefeasibility of primal variables, nonlinear Lagrange methods are playing important rolesin solving constrained optimization problems. This desseration attempts to extend thismethod to solving NCSDP for the proficiency of nonlinear lagrangeian alogrithm solv-ing nonlinear programming and extensive utilization of NCSDP in practical problems.we construct a dual alogrithm basing on a nonlinear lagrangian for solving NCSDP andthen prove its local convergence.This is to say,the sequence producted by the alogrithmconverge to KKT points of primal problem. And at last we establish the error estimateformula, the main results,obtained in this disseration,may be summerized as follows:1.Chapter2 is devoted to summarizing the optimal conditions of NCSDP which areindispensable to the next chapter.This chapter at first introduces the optimal conditionsof abstract constraint optimization.And then utilizes those conditions to the NCSDP.2.Chapter3 is devoted to bringing forward a nonlinear Lagrange algorithm for NCSDPprogramming and proving its convergence.Namely, under some mild assumptions,it is provedthat there exists a threshold of the penalty parameter such that these sequences gener-ated by the nonlinear algorithm locally converge to the KKT point of NCSDP when thepenalty parameter is less than the threshold. Moreover, we establish the error bound ofthe solutions with the penalty parameter.更多还原