Eigenvalue Decomposition Based Modified Newton Algorithm

When the Hessian matrix is not positive, the Newton direction maybe not the descending direction. A new method named eigenvalue decomposition based modified Newton algorithm is presented, which first takes eigenvalue decomposition on the Hessian matrix, then replaces the negative eigenvalues with their absolutely values, finally reconstruct Hessian matrix and modify searching direction. The new searching direction is always the descending direction, and the convergence of the algorithm is proved and conclusion on convergence rate is presented qualitatively. At last, a numerical experiment is given for comparing the convergence domains of modified algorithm and classical algorithm.