A tensor trust-region model for nonlinear system

It has turned out that the tensor expansion model has better approximation to the objective function than models of the normal second Taylor expansion. This paper conducts a study of the tensor model for nonlinear equations and it includes the following: (i) a three dimensional symmetric tensor trust-region subproblem model of the nonlinear equations is presented; (ii) the three dimensional symmetric tensor is replaced by interpolating function and gradient values from the most recent past iterate, which avoids the storage of the three dimensional symmetric tensor and decreases the workload of the computer; (iii) the limited BFGS quasi-Newton update is used instead of the second Jacobian matrix, which generates an inexpensive computation of a complex system; (iv) the global convergence is proved under suitable conditions. Numerical experiments are done to show that this proposed algorithm is competitive with the normal algorithm.


Introduction
This paper focuses on S(x) = 0, x ∈ n , (1.1) where S : n → n is continuously differentiable nonlinear system. The nonlinear system (1.1) has been proved to possess wildly different application fields in parameter estimating, function approximating, and nonlinear fitting, etc. At present, there exist many effective algorithms working in it, such as the traditional Gauss-Newton method [1, 9-11, 14, 16], the BFGS method [8,23,27,29,39,43], the Levenberg-Marquardt method [6,24,42], the trust-region method [4,26,35,41], the conjugate gradient algorithm [12,25,30,38,40], and the limited BFGS method [13,28]. Here and in the next statement, for research convenience, suppose that S(x) has solution x * . Setting β(x) := 1 2 S(x) 2 as a norm function, the problem (1.1) is equivalent to the following optimization problem: (1. 2) The trust-region (TR) methods have as a main objective solving the so-called trustregion subproblem model to get the trial step d k , where x k is the kth iteration, is the so-called TR radius, and · is the normally Euclidean norm of vectors or matrix. The first choice for many scholars is to study the above model to make a good improvement. An adaptive TR model is designed by Zhang and Wang [42]: where p > 0 is an integer, and 0 < c < 1 and 0.5 < γ < 1 are constants. Its superlinear convergence is obtained under the local error bound assumption, by which it has been proved that the local error bound assumption is weaker than the nondegeneracy [24]. Thus one made progress in theory. However, its global convergence still needs the nondegeneracy.
Another adaptive TR subproblem is defined by Yuan et al. [35]: where B k is generated by the BFGS quasi-Newton formula is the next iteration, and B 0 is an initial symmetric positive definite matrix. This TR method can possess the global convergence without the nondegeneracy, which shows that this paper made a further progress in theory. Furthermore, it also possesses the quadratic convergence. It has been showed that the BFGS quasi-Newton update is very effective for optimization problems (see [32,33,36] etc.). There exist many applications of the TR methods (see [19][20][21]31] etc.) for nonsmooth optimizations and other problems.
It is not difficult to see that the above models only get the second Taylor expansion and approximation. Can we get the approximation to reach one more level, namely the third expansion, or even the fourth? The answer is positive and a third Taylor expansion is used and a three dimensional symmetric tensor model is stated. In the next section, the motivation and the tensor TR model are stated. The algorithm and its global convergence are presented in Sect. 3. In Sect. 4, we do the experiments of the algorithms. One conclusion is given in the last section.

Motivation and the tensor trust-region model
Consider the tensor model for the nonlinear system S(x) at x k , where ∇S(x k ) is the Jacobian matrix of S(x) at x k and T k is three dimensional symmetric tensor. It is not difficult to see that the above tensor model (2.1) has better approximation than the normal quadratical trust-region model. It has been proved that the tensor is significantly simpler when only information from one past iterate is used (see [3] for details), which obviously decreases the complexity of the computation of the three dimensional symmetric tensor T k . Then the model (2.1) can be written as the following extension: In order to avoid the exact Jacobian matrix ∇S(x k ), we use the quasi-Newton update matrix B k instead of it. Thus, our trust-region subproblem model is designed by where B k = H -1 k and H k is generated by the following low-storage limited BFGS (L-BFGS) update formula: I is the unit matrix and m is a positive integer. It has turned out that the L-BFGS method has a fast linear convergence rate and minimal storage, and it is effective for large-scale problems (see [2,13,28,34,37] etc.). Let d p k be the solution of (2.3) corresponding to the constant p. Define the actual reduction by (2.5) and the predict reduction by Based on definition of the actual reduction Ad k (d p k ) and the predict reduction Pd k (d p k ), their radio is defined by . (2.7) Therefore, the tensor trust-region model algorithm for solve (1.1) is stated as follows.
Remark The procedure of "Step 2-Step 3-Step 2" is called the inner cycle in the above algorithm. It is necessary for us to prove that the inner cycle is finite, which generates the circumstance that Algorithm 1 is well defined.

Convergence results
This section focuses on convergence results of Algorithm 1 under the following assumptions.

Assumption i (A)
The level set Ω defined by holds.
Proof By the definition of d p k of (2.3), then, for any α ∈ [0, 1], we get Therefore, we have The proof is complete. Proof Using Assumption i, the definition of (2.5) and (2.6), we obtain This completes the proof. Proof This lemma will be proved by contradiction. Suppose, at x k , that Algorithm 1 infinitely circles in the inner cycle, namely, r p k < ρ and c p → 0 with p → ∞. This implies that g k ≥ , or the algorithm stops. Thus we conclude that d p k ≤ k = c p g k → 0 is true. By Lemma 3.1 and Lemma 3.2, we get Therefore, for p sufficiently large, we have which generates a contradiction with the fact r p k < ρ. The proof is complete.

Lemma 3.4 Suppose that the conditions of Lemma 3.3 holds. Then we conclude that {x k } ⊂ Ω is true and {β(x k )} converges.
Proof By the results of the above lemma, we get Combining with Lemma 3.1 generates Then {x k } ⊂ Ω holds. By the case β(x k ) ≥ 0, we deduce that {β(x k )} converges. This completes its proof. holds. Using (3.3) one gets (3.8). So, we can complete this lemma by (3.9). We use the contradiction to have (3.9). Namely, we suppose that there exist an subsequence {k j } and a positive constant ε such that (3.10) Let K = {k | B k S(x k ) ≥ ε} be an index set. Using Assumption i, the case B k S(x k ) ≥ ε (k ∈ K ), and S(x k ) (k ∈ K ) is bounded away from 0, we assume holds. By Lemma 3.1 and the definition of Algorithm 1, we obtain where p k is the largest p value obtained in the inner circle. Lemma 3.4 tells us that the sequence {β(x k )} is convergent, thus Then p k → +∞ when k → +∞ and k ∈ K . Therefore, for all k ∈ K , it is reasonable for us to assume p k ≥ 1. In the inner circle, by the determination of p k (k ∈ K ), let d k corresponding to the subproblem Using Lemma 3.1 and the definition k one has Using Lemma 3.2 one gets Thus, we obtain Using p k → +∞ when k → +∞ and k ∈ K , we get this generates a contradiction to (3.12). This completes the proof.

Numerical results
This section reports some numerical results of Algorithm 1 and the algorithm of [35] (Algorithm YL).

Problems
The nonlinear system obeys the following statement:

Problem 10
The discretized two-point boundary value problem similar to the problem in [17] S(x) = Ax + 1 (n + 1) 2 F(x) = 0, with A is the n × n tridiagonal matrix given by

Results and discussion
The column meaning of the tables is as follows. Dim: the dimension. NI: the iterations number. NG: the norm function number. Time: the CPU-time in s. Numerical results of Table 1 show the performance of these two algorithms as regards NI, NG and Time. It is not difficult to see that: (i) Both of these algorithms can successfully solve all these ten nonlinear problems; (ii) the NI and the NG of these two algorithm do not increase when the dimension becomes large; (iii) the NI and the NG of Algorithm 1 are competitive to those of Algorithm YL and the Time of Algorithm YL is better than that of Algorithm 1. To directly show their the efficiency, the tool of [5] is used and three figures for NI, NG and Time are listed. Figures 1-3 show the performance of NI, NG and Time of these two algorithms. It is easy to see that the NI and the NG of Algortihm 1 have won since their performance profile plot is on top right. And the Time of Algorithm YL has superiority to Algorithm 1. Both of these two algorithms have good robustness. All these three figures show that both of these two algorithms are very interesting and we hope they will be further studied in the future.

Conclusions
This paper considers the tensor trust-region model for nonlinear system. The global convergence is obtained under suitable conditions and numerical experiments are reported. This paper includes the following main work: (1) a tensor trust-region model is established and discussed.
(2) the low workload update is used in this tensor trust-region model. In the future, we think this tensor trust-region model shall be more significant.