Received 17 January 2016; accepted 12 March 2016; published 15 March 2016
We consider the unconstrained optimization problem  -
where is twice continuously differentiable, whose gradient and Hessian are denoted by and respectively. Throughout this paper, we assume that the solution set of (1.1) is nonempty and denoted by X, and in all cases refers to the 2-norm.
It is well known that is convex if and only if is symmetric positive semidefinite for all. Moreover, if is convex, then if and only if x is a solution of the system of nonlinear equations
Hence, we could get the minimizer of by solving (1.2)  - . The Newton method is one of efficient solution methods. At every iteration, it computes the trial step
where and. As we know, if is Lipschitz continuous and nonsingular at the solution, then the Newton method has quadratic convergence. However, this method has an obvious disadvantage when the is singular or near singular. In this case, we may compute the Moore-Penrose step  . But the computation of the singular value decomposition to obtain is sometimes prohibitive. Hence, computing a direction that is close to may be a good idea.
To overcome the difficulty caused by the possible singularity of,  proposed a regularized Newton method, where the trial step is the solution of the linear equations
where I is the identity matrix. is a positive parameter which is updated from iteration to iteration.
Now we need to consider another question, “how to choose the regularized parameter?” Yamashita and
Fukushima  chose and showed that the regularized Newton method has quadratic convergence under the local error bound condition which is weaker than nonsingularity. Fan and Yuan  took
with and showed that the Levenberg-Marqulardt method preserves the quadratic convergence. Numerial results (  ) show that the choice of performs more stable and preferable.
Inspired by the regularized Newton method  with correction for nonlinear equations, we propose a modified regularized Newton method in this paper. At every iteration, the modified regularized Newton method first solves the linear equations
to obtain the Newton step, where is updated from iteration to iteration, and solves the linear equations
to obtain the approximate Newton step.
It is easy to see
Then it solves the linear equations
to obtain the approximate Newton step.
The aim of this paper is to study the convergence properties of the above modified regularized Newton method and do a numerical experiment to test its efficiency.
The paper is organized as follows. In Section 2, we present a new regularized Newton algorithm with correction by trust region technique, and then prove the global convergence of the new algorithm under some suitable conditions. In Section 3, we test the regularized Newton algorithm with correction and compared it with a regularized Newton method. Finally, we conclude the paper in Section 4.
2. The Algorithm and Its Global Convergence
In this section, we first present the new modified regularized Newton algorithm by using trust region technique, then prove the global convergence. First, we give the modified regularized Newton algorithm.
Let and be given by (1.6) and (1.8), respectively. Since the matrix is symmetric and positive definite, is a descent direction of at, however may not be. Hence we prefer to use a trust region technique.
Define the actual reduction of at the kth iteration as
Note that the regularization step is the minimizer of the convex minimization problem
If we let
then it can be proved  that is also a solution of the trust region problem
By the famous result given by Powell in  , we know that
By some simple calculations, we deduce from (1.7) that
so, we have
Similar to, is not only the minimizer of the problem
but also a solution to the trust region problem
where. Therefore we also have
Based on the inequalities (2.2), (2.3) and (2.4), it is reasonable for us to define the new predicted reduction as
The ratio of the actual reduction to the predicted reduction
plays a key role in deciding whether to accept the trial step and how to adjust the regularized parameter.
The regularized Newton algorithm with correction for unconstrained convex optimization problems is stated as follows.
Step 1. Given, , ,. Set.
Step 2. If, then stop.
Step 3. Compute.
to obtain and set
to obtain and set
Step 4. Compute. Set
Step 5. Choose as
Set and go step 2.
Before discussing the global convergence of the algorithm above, we make the following assumption.
Assumption 2.1. and are both Lipschitz continuous, that is, there exists a constant, such that
It follows from (2.14) that
The following lemma given below shows the relationship between the positive semidefinite matrix and symmetric positive semidefinite matrix.
Lemma 2.1. A real-valued matrix A is positive semidefinite if and only if is positive semidefinite.
Proof. See  . ♢
Next, we give the bounds of a positive definite matrix and its inverse.
Lemma 2.2. Suppose A is positive semidefinite. Then,
hold for any.
Proof. See  . ♢
Theorem 2.1. Under the conditions of Assumption 2.1, if f is bounded below, then Algorithm 2.1 terminates in finite iterations or satisfies
Proof. We prove by contradiction. If the theorem is not true, then there exists a positive and an integer such that
Without loss of generality, we can suppose. Set. Then
Now we will analysis in two cases whether T is finite or not.
Case (1): T is finite. Then there exists an integer such that
By (2.11), we have
Therefore by (2.12) and (2.17), we deduce
Since, , we get from (2.8) and (2.18) that
Duo to (1.7), we get
From (2.10), we obtain
where is a positive constant.
It follows from (2.1) and (2.5) that
Moreover, from (2.6), (2.17), (2.13) and (2.19), we have
for sufficiently large k.
Duo to (2.21) and (2.22), we get
which implies that. Hence, there exists positive constant such that, holds for all large k, which contradicts to (2.18).
Case (2): T is infinite. Then we have from (2.6) and (2.17) that
which implies that
The above equality together with the updating rule of (2.12) means
Similar to (2.20), it follows from (2.25) and (2.26) that
for some positive constant. Then we have
This equality together with (2.24) yields
which implies that
It follows from (2.8), (2.27), (2.26) and (2.20) that
Since from (2.8), we have from (2.13), (2.17) and (2.28) that
By the same analysis as (2.23) we know that
Hence, there exists a positive constant such that holds for all sufficiently large k, which gives a contradiction to (2.29). The proof is completed. ♢
3. Numerical Experiments
In this section, we test the performance of Algorithm 2.1 on the unconstrained nonlinear optimization problem, and compared it with a regularized Newton method without correction. The function to be minimized is
where are constants. It is clear that function is convex and the minimizer set of is
The Hessian is given by
where,. Matrix is positive semidefinite for all x, but
singular as the sum of every column is zero. Since the Hessian is always singular, the Newton method cannot be used to solve nonlinear Equations (1.2). But by using the regularization technique, both regularized Newton method and Algorithm 2.1 work quite well.
The aims of the experiments are as follows: to check whether Algorithm 2.1 converges quadratically as stated in Section 3 and also to see how well the technique of correction works. We set, , , , , and for Algorithm 2.1.
Table 1 reports the norms of at every iteration when, ,
and. Algorithm 2.1 only take four iterations to obtain the minimizer of; decreases very quickly. The results show the sequence quadratic convergence. The iteration is as follows
We may observe that the whole sequence converges to.
We also ran the regularized Newton algorithm (RNA) without correction, that is, we do not solve the linear equations (2.9)-(2.10) and just set the solution of (2.8) to be the trial step. Then, we tested the regularized Newton algorithm without correction and Algorithm 2.1 for various of n, and different choices of the starting point. The results are listed in Table 2.: the selected value of; Dim: the dimension n of the problem;: the ith element; niter: the number of iterations required;: the final value of;: the final value of. We use as the stopping criterion.
Table 1. Results of Algorithm 2.1 to test quadratic convergence.
Table 2. Results of RNA and Algorithm 2.1.
Moreover, we can see for the same, n and, the number of iterations of Algorithm 2.1 is always less than that of RNA. And the correction term does help to improve RNA when the initial point is far away from the minimizer. These facts indicate that the introduction of correction is really useful and could accelerate the convergence of the regularized Newton method.
4. Concluding Remarks
In this paper, we propose a regularized Newton method with correction for unconstrained convex optimization. At every iteration, not only a RNM step is computed but also two correction steps are computed which make use of the previous available Jacobian instead of computing the new Jacobian. Numerical experiments suggest that the introduction of correction is really useful.
This research is supported by the National Natural Science Foundation of China (11426155) and the Hujiang Foundation of China (B14005).