Newton Type Algorithms For Singular Convex Optimization Problems And Constrained Monotone Nonlinear Equations

Newton methods are a class of efficient algorithms for solving nonlinear equations and optimization problems, which converge very fast with high precision. If the Jacobian or Hessian at the solution is nonsingular and the Jacobian or Hessian are Lipschitz continuous near the solution, then the classical Newton method has quadratic convergence rate. However, the nonsingularity is a strong assumption condition, which implies that the problem has a unique local solution. This paper is devoted to investigating the global and local convergence properties of Newton type algorithms for some nonlinear equations and optimization problems.In Chapter1, we simply introduce the background of the problems which we will study and provide some preliminary data.In Chapter2, we discuss the unconstrained convex optimization problem with singular solutions. We present a modified regularized Newton method for this problem and show its global and quadratic convergence under the local error bound condition. Furthermore, we obtain its cubic convergence by the use of the singular value decomposition of matrix under the same assumptions.In Chapter3, we mainly study Newton type algorithms for solving the constrained monotone nonlinear equations. We propose a projection regularized Newton method for this problem and prove that the proposed method has global and locally quadratic convergence properties under the local error bound condition which is weaker than the nonsingularity condition. This result does hold whether the solution set is unique or not. In Chapter4, we did some numerical experiments and reported some limited numerical results, which show that the proposed algorithm in Chapter3is efficient.