New Spectral Conjugate Gradient Algorithms And Applications

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...