【作者】 邓松海；

【导师】 万中；

【作者基本信息】 中南大学 ， 运筹学与控制论， 2013， 博士

【摘要】 谱共轭梯度法是近年来引起人们广泛兴趣的一类新型无约束优化数值方法.它以共轭梯度法为基础,又和谱方法具有一定共性.本文研究了若干谱共轭梯度法,研究内容包括用于产生搜索方向的谱参数和共轭参数的选取问题,确定线搜索策略的问题,全局收敛性的建立,及其在管理科学中的应用等.我们首先在第2章解决了一类PRP谱共轭梯度法的全局收敛性.用数值实验的方法研究了先前由万中等人提出的方法中用于建立全局收敛性的一个假设条件.结果证实该条件不是总成立.本文的工作就是去掉了原有的假设条件,而又不破坏原有的全局收敛性结论,重新建立了全局收敛性定理,从而保证了该方法具有更广泛的应用范围.其次,我们在第3章提出了一类新型的Dai-Liao(DL)型共轭梯度法.该方法不仅是DL方法的一个推广,而且提出了一种改进的线搜索：通过实施该线搜索过程,自动得出下一步方向的共轭参数值.我们证明了由本文提出的方法所得到的方向一定是充分下降方向,证明了新型DL算法是适定的,是全局收敛的,数值实验表明新型DL方法比原方法效率有显著的提高.第4章提出了一类增强的谱共轭梯度法.利用搜索方向尽可能向牛顿方向靠近的性质,选取谱参数和共轭参数,得到的搜索方向具有充分下降性,且具备牛顿方向的特征.该方法推广了N. Andrei的思想,在算法设计或收敛性证明中我们去掉了N. Andrei关于算法参数的一个假设条件.在一种Armijo型线搜索下,证明了算法的全局收敛性.与公认的高效算法CG_DESCENT, SCALCQ AMDYN等作比较,数值实验证明本章提出的方法的数值行为优越.以增强的谱共轭梯度法为基础,我们在第5章提出了拟牛顿共轭梯度法,它是基于拟牛顿法构建谱系数的新方法.为了使本方法具有共轭梯度所具备的内存占用少、效率高的特点,这里的拟牛顿法产生的矩阵是对角稀疏矩阵.这种对角拟牛顿共轭梯度法既具有拟牛顿法的高时效性,又有共轭梯度法具有的低存贮性,适合于解决大规模无约束优化问题.数值实验表明,该方法比单独的对角拟牛顿法和单独的共轭梯度法迭代次数都少.最后,针对现有文献在假定多产品客户需求是相互独立的随机变量的条件下研究报童问题之不足,我们用三种Copula模拟建立两产品报童问题的两产品需求非独立时的联合分布函数.用指数效用函数建立了风险厌恶的期望和产品订货量之间的二元函数关系,用谱共轭梯度方法求解.本文还首次用数值计算的方法证实了两产品的相关性与风险的有关结论.图5幅,表10个,参更多还原

【Abstract】 Recently, the spectral conjugate gradient methods have been attracting an extensive interest in the research field of unconstrained numerical optimization. These methods are based on conjugate gradient methods as well as closely relating with the spectral gradient methods.In this dissertation, we intend to study some types of spectral conjugate gradient methods. Our focuses are on the choices of spectral parameter and conjugate parameter to generate suitable search directions, the strategies of line search, the establishment of global convergence, and the application of the developed algorithms in management science.In Chapter2, we first investigate the global convergence for a class of PRP spectral conjugate gradient method. By numerical experiments, an assumption condition is checked, which is used to establish the global convergence of an algorithm developed by Wan etc. in their earlier article. The assumption is proved not always true. The work of this paper is removing the old assumption without destroying the global convergence and rebuilding new global convergence theorem. This modification greatly expands the scope of application of Wan’s method.Next, in Chapter3, a new Dai-Liao (DL) type conjugate gradient method is proposed. This method is an extension of DL as well as including an improved strategy of line search. By the new line search, a conjugate parameter for the next search direction is obtained automatically. We prove that the search direction generated by our method is a sufficient direction and that the new type DL is well-defined, global convergent. The numerical experiment shows that the new type DL is significantly more efficient than the old one.In Chapter4, an improved spectral conjugate gradient method is proposed. In this method, the spectral parameter and the conjugate parameter are chosen simultaneously such that the search direction at each iteration is close to Newton direction as well as being sufficiently descent. The new method has the features of Newton direction, generalizes the thinking of N. Andrei and removes its extral assumption condition. Compared with the state-of-the-art algorithms CG_DESCENT, SCALCQ AMDYN, the improved spectral conjugate gradient method performs better.On the basis of the improved conjugate gradient method proposed in Chapter4, a diagonal quasi Newton conjugate gradient method is investigated in Chapter5, where the spectral parameter is obtained by a quasi-Newton type method. In order to make it have less memory ocuppation and high efficiency as conjugate gradient methods do, the matrix obtained by quasi-Newton is sparse. The new diagonal quasi-Newton conjugate gradient method not only has the high efficiency of quasi-Newton mehtod but also has less memory occupation of conjugate gradient method. Therefore it suits for solving large-scale unconstrained optimization problems. Numerical experiement shows that the iteration times is less than that of diagonal quasi-Newton and that of conjugate gradient, respectively.At last, aiming at the shortage in multi-product newsboy model that the customer demands in the existent literature are assumed to be multually independent random variables, we construct joint culmulative distribution function of independent customer demands of two-product newsvendor problem by using three types of Copulas. By employing exponential utility function, we obtain the two variate functions between utility expectation of profit and quantity of product order under the risk averse cases. After analysizing the analytical properties of expectation of utility profit, we apply our proposed spectral conjugate gradient method to solve the minimum problem of objective function. It is shown that the reliability that about two-product relevance and profit is true:the greater is the relevance of two-product, the more is the expected profit The paper is attached with5figures,10更多还原