Unconstrained Optimization Problems Newton Filter Algorithm

In this paper, by introducing a filter in Newton's method, we propose a Newton-filter method for solving unconstrained optimizatin problems.The filter technique was first studied by Fletcher and Leyffer in the sequential quadratic programming (SQP) method for solving constrained optimization problems. The purpose of the SQP-filter method is to overcome the numerical difficulty caused by the use of large penalty factor in the SQP method. The basic idea of an SQP-filter method is that at each step, the method generates a point such that either the value of the objective function descreases sufficiently, or the deterioration of the constraints descreases sufficiently, or both. Consequently, in an SQP-filter method, the use of penalty function is avoid. Under certain conditions, the SQP-filter method is proved to be globally convergent. Some numerical experiments show that the method is practically efficient. However, so far, thereis no theory about the convergence rate of the filter method. It is also unknown whether the unit step can be accepted.In this paper, we adopt the idea of SQP-filter to develop a Newton-filter method. Newton method is favorable for the reason that it has two-convergent ratio under some conditions. Observing that the Newton direction is always a descent direction of the square norm of the gradient, even if it is not a descent direction of the objective function, which will result some linear research-such as Armijo research or Wolfe research-fail. Based on this observation, we develop a Newton-filter method in such a way that at each step, the method generates a point such that either the objective function value or its gradient descreases sufficiently. Under appropriate conditions, we obtain the global and quadratic convergence of the proposed method. Moreover, we show that after finite iterations, the unit step is always accepted.