Based On Cauchy's Second-order Cheng Diagonal Quasi-newton Algorithm Research

The optimization is the selection of the best projects in many kinds offeasible projects, with the rapid development of science and technology, the opt-imization is widely used in agriculture, engineering design, modern management,transportation, national defense and other important fields. The Quasi-Newtonmethod is one of the most effective ways for solving the unconstrained problems,but there is the matrix which is stored in the Quasi-Newton method. So dealingwith the large scale problems, we need quite large storing space andcomputation. The diagonal Quasi-Newton method can reduce the computationand storing space, so that it is suitable for solving the large scale optimizationproblems. Diagonal Quasi-Newton method based on second order Quasi-Cauchyequation is a hot spot in recent study, of which diagonal Quasi-Cauchy methodand diagonal third order Quasi-Cauchy method are proposed, which provides anew train of thought for solving large scale unconstrained optimizationproblems.Similar to Quasi-Cauchy equation, in the thesis we propose the second orderQuasi-Cauchy equation for the first time and construct a diagonal updatingformula based on the second order Quasi-Cauchy equation and the least-changediagonal updating strategy. On the basis of the diagonal updating formula basedon the second order Quasi-Cauchy equation, we also propose two kinds ofsynchronous diagonal Quasi-Newton methods and an asynchronous diagonalQuasi-Newton method. In the thesis we analyze the convergence of the threealgorithms and make some numerical experiments for the three algorithms.In chapter2, based on second order Quasi-Cauchy equation, we propose adiagonal updating formula and study the sufficient conditions and necessaryconditions of making the updating formula positive definite.In chapter3and chapter4, based on second order Quasi-Cauchy equation,we propose two kinds of synchronous diagonal Quasi-Newton methods andstudy the convergence of the two algorithms. In the synchronous diagonalQuasi-Newton methods the updating matrix is replaced as a whole; synchronous diagonal Quasi-Newton methodⅡis the improvement of synchronous diagonalQuasi-Newton methodⅠ; the convergence conditions of synchronous diagonalQuasi-Newton methodⅡis relatively weak. Numerical results also show that thesynchronous diagonal Quasi-Newton methodⅠandⅡare efficient to go a stepfurther and they are suitable for solving large-scale unconstrained optimizationproblems.In chapter5, based on second order Quasi-Cauchy equation, we propose anasynchronous diagonal Quasi-Newton method and study the convergence of thealgorithm. In the algorithm the updating matrix is replaced locally. Underappropriate assumptions, we also prove the linear convergence of the algorithmto go a step further. Numerical results also show that the algorithm is effectiveand practical.