Abstract
In this paper, we present a modified regularized Newton method for minimizing a nonconvex function whose Hessian matrix may be singular. We show that if the gradient and Hessian of the objective function are Lipschitz continuous, then the method has a global convergence property. Under the local error bound condition which is weaker than nonsingularity, the method has cubic convergence.
Highlights
IntroductionX∈Rn where f : Rn → R is twice continuously differentiable, whose gradient ∇ f and Hessian ∇2 f are denoted by g(x) and H(x) respectively
We consider the unconstrained optimization problem min f (x), (1)x∈Rn where f : Rn → R is twice continuously differentiable, whose gradient ∇ f and Hessian ∇2 f are denoted by g(x) and H(x) respectively
We present a modified regularized Newton method for minimizing a nonconvex function whose Hessian matrix may be singular
Summary
X∈Rn where f : Rn → R is twice continuously differentiable, whose gradient ∇ f and Hessian ∇2 f are denoted by g(x) and H(x) respectively. If Hk is Lipschitz continuous and nonsingular at the solution, the Newton method has quadratic convergence. Sun, 1999) proposed a regularized Newton method, where the trial step is the solution of the linear equations (Hk + μkI)d = −gk,. H. Li, 2004) chose μk = ∥gk∥2 and showed that the regularized Newton method has quadratic convergence under the local error bound condition which is weaker than nonsingularity. Newton and inexact Newton methods are possible extended to nonconvex minimization problems (Dong-huiLi, 2004) The main scheme of the modified regularized Newton method for unconstrained nonconvex optimization is given as follows. At every iteration, it solves the linear equations (Hk + μkI)d = −gk (7).
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