A Modified Tensor Method For Singular Unconstrained Optimization

The tensor method, which is a generalization of the standard Newton model, was proposedby Schnabel and Frank in1984. This method uses the four-order term of the Taylor expansion, fixup the weakness that the Newton model will lose the fast local convergence rate when the Hessianis singular at the minimizer, and show the advantages that it can solve the singular problem.In the thesis we mainly discuss the algorithms and theory of tensor methods which cansolve the singular unconstrained optimization problems. Rather than with the difference offunctions and gradients, the modified tensor model is constructed with the difference of gradientsand Hessian. The modified tensor method is proposed, and the numerical results are reported.The structure of this paper is organized as follows. The first chapter describes the originsand progress in research of tensor method for solving unconstrained optimization. The secondchapter describes some basic knowledge of this article, the theoretical knowledge and algorithmsof tensor method and the linear search method. In the third chapter, we propose a modified tensormodel for singular unconstrained optimization, and describe the algorithm of this modified tensormodel for singular unconstrained problems. In the forth chapter, the numerical results of themodified and standard tensor methods are reported and compared. Finally, the conclusion is given.