A new Newton method for convex optimization problems with singular Hessian matrices

In this paper, we propose a new Newton method for minimizing convex optimization problems with singular Hessian matrices including the special case that the Hessian matrix of the objective function is singular at any iteration point. The new method we proposed has some updates in the regularized parameter and the search direction. The step size of our method can be obtained by using Armijo backtracking line search. We also prove that the new method has global convergence. Some numerical experimental results show that the new method performs well for solving convex optimization problems whose Hessian matrices of the objective functions are singular everywhere.