Unconstrained Optimization Problems Diagonal Third-order Correction Algorithm

The optimization problem is the selection of the best decisions in a limitedor unlimited kinds of decision-making, it has wide use in many fields, such as,economics, engineering, modern management, transportation and nationaldefense. In recent years, the quasi-Newton method is being paid more and moreattention. However, the application of the method for large scale problemrequires vast computational resource to form and store updated matrix. In thisthesis, we focus on the diagonal updated algorithms which can reduce theamount of storage and computation significantly,the main results are as follows:In chapter1, some optimization algorithms,basic knowledge of thediagonal updated method and the main work of the paper are introduced.In chapter2, we propose a diagonal tree-order quasi-Newton method forlarge scale unconstrained optimization problems. The inverse of Hessenapproximation in diagonal matrix form can be obtained, while avoiding thecomputational and storing expenses of actual calculation. The globalconvergence and the superlinear convergence property are analyzed. Numericalresults show that this method is efficient.In chapter3, we propose a new gradient-type method for solvingunconstrained optimization problems, which is named diagonal tree-orderquasi-Cauchy method. A simple monotone strategy is presented in the frame ofBarzilai and Borwein(BB) method. The new method uses approximation of theHessen in diagonal matrix form based on the three-order quasi-Cauchy equationrather than the multiple of the identity matrix in the BB method. Byincorporating a simple monotone strategy, the approach belongs to descentalgorithm, and the linearly convergence is achieved. Numerical experimentsshow that the proposed method yields desirable improvement.