Modified Conjugate Gradient Method And Its Global Convergence

This thesis mainly analyzes the conjugate gradient method, which is very effective insolving large-scale unconstrained optimization. The global convergences of the modifiedDY method, the modified HS method as well as the PRP method, the CD method and theLS method, which are under the new two nonmonotone Armijo type line searches, arediscussed separately. The results extend other people's conclusions.In the first chapter, on the base of the usual DY and HS formulas, we propose themodified DY and HS formulas separately and apply them into unconstrained optimizationproblems to get new kind of conjugate gradient algorithms. On the condition ofnonnegative constraining, the new formulas can keep the sufficiently descent propertywithout any line searches. Under some condition, algorithms combining the new formulasand any line searches which can ensure the Zoutendijk condition hold, have the globalconvergent property. Further, under some faintish condition, the new formulas convergentglobally with the Armijo line search.In the second chapter, two new line search rules are proposed. Nonmonotone linesearch algorithms have potential in solving large-scale nonlinear optimization problems.For some cases, they are more effective than those monotone algorithms. In recent years,the Armijo type line searches are often used in those literatures studying conjugategradient methods. In this section, two kinds of new nonmonotone Armijo type linesearches are proposed combining the advantages of the nonmonotone line searches andthe Armijo type line searches. Their feasibilities are proved and their basic properties aregiven by two lemmas.In the third chapter, the conjugate gradient methods under two nonmonotone Armijotype line searches are studied. Under this kind of line searches, the PRP method, the CDmethod and the LS method all can keep the sufficient descent property. The newconjugate gradient algorithms are proposed and their global convergences are proved.