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Global Convergence Properties Of Conjugate Gradient Methods

Posted on:2005-03-31Degree:DoctorType:Dissertation
Country:ChinaCandidate:S J LianFull Text:PDF
GTID:1100360122996892Subject:Operational Research and Cybernetics
Abstract/Summary:PDF Full Text Request
The global convergence properties of the conjugate gradient methods for the unconstrained optimization problems are studied in this dissertation. The aim of this dissertation is to give some theoretical analysis.Chapter 1 is the introduction of this dissertation, which introduces nonlinear conjugate gradient methods, the context of this dissertation and the main results obtained in this dissertation.In Chapter 2, the author proposes two new dependent Fletcher-Reeves conjugate gradient methods arising from different choice for the scalarβk and make two different kinds ofestimations of upper bounds of \βk \ with respect to βkFR, which are based on Abel Theorem of non-convergent series of positive items. With several different line searches, global convergence results are established for the two new methods, which extend the previous dependent Fletcher-Reeves conjugate gradient methods.A new region of βk with respect to βkPRP is given in Chapter 3. With two Armijo-type line searches, the author investigates the global convergence properties of the dependent PRP conjugate gradient methods, which extend the global convergence results of PRP conjugate gradient method proved by Grippo and Lucidi (1997) and Dai and Yuan (2002). With the Wolfe line search, the global convergence of the conjugate gradient methods with the negative βk is obtained. Further, the global convergence results of the dependent PRP and HS conjugate gradient methods are proved using the Wolfe line search.In Chapter 4, two new Armijo-type line searches are proposed. One of which simplifies greatly the Armijo-type line search given by Grippo and Lucidi (1997). This chapter investigates global convergence properties of the conjugate descent (CD) method with the two Armijo-type line searches. The CD method can ensure all descent search directions satisfy the sufficient descent condition.In Chapter 5, the author investigates the global convergence properties of the FR and PRP conjugate gradient methods using the two Armijo-type line searches proposed in Chapter 4. Further, the author investigates the global convergence of the conjugate gradient methods with the negative βk under the two Armijo-type line searches.In Chapter 6, the two Armijo-type line searches proposed in Chapter 4 are shown toguarantee the global convergence of the DY method for the unconstrained minimization of nonconvex differentiable functions. Further, if the function is strictly convex, the two Armijo-type line searches and another Armijo-type line search are also shown to guarantee the convergence of the DY method.In Chapter 7, the author investigates a class of conjugate gradient methods, which can be regarded as some kind of convex combination of the Fletcher-Reeves method and the method proposed by Dai et al. The two Armijo-type line searches proposed in Chapter 4 are shown to guarantee the global convergence of the class of conjugate gradient methods.
Keywords/Search Tags:conjugate gradient method,unconstrained optimization, line search,global convergence,descent direction
PDF Full Text Request
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