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A tensor trustregion model for nonlinear system
Journal of Inequalities and Applications volume 2018, Article number: 343 (2018)
Abstract
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 trustregion 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 quasiNewton 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
where \(S:\Re^{n} \rightarrow \Re^{n}\) 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, 9,10,11, 14, 16], the BFGS method [8, 23, 27, 29, 39, 43], the Levenberg–Marquardt method [6, 24, 42], the trustregion 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 \(\beta (x):=\frac{1}{2}\Vert S(x) \Vert ^{2}\) as a norm function, the problem (1.1) is equivalent to the following optimization problem:
The trustregion (TR) methods have as a main objective solving the socalled trustregion subproblem model to get the trial step \(d_{k}\),
where \(x_{k}\) is the kth iteration, △ is the socalled TR radius, and \(\Vert \cdot \Vert \) 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 \(0.5<\gamma <1\) 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 quasiNewton formula
where \(y_{k}=S(x_{k+1})S(x_{k}),\,s_{k}=x_{k+1}x_{k}\), \(x_{k+1}\) 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 quasiNewton 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 trustregion model
Consider the tensor model for the nonlinear system \(S(x)\) at \(x_{k}\),
where \(\nabla S(x_{k})\) 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 trustregion 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 \(\nabla S(x_{k})\), we use the quasiNewton update matrix \(B_{k}\) instead of it. Thus, our trustregion subproblem model is designed by
where \(B_{k}=H_{k}^{1}\) and \(H_{k}\) is generated by the following lowstorage limited BFGS (LBFGS) update formula:
where \(\rho_{k}=\frac{1}{s_{k}^{T}y_{k}}\), \(V_{k}=I\rho_{k}y_{k}s _{k}^{T}\), I is the unit matrix and m is a positive integer. It has turned out that the LBFGS method has a fast linear convergence rate and minimal storage, and it is effective for largescale problems (see [2, 13, 28, 34, 37] etc.). Let \(d_{k}^{p}\) be the solution of (2.3) corresponding to the constant p. Define the actual reduction by
and the predict reduction by
Based on definition of the actual reduction \(Ad_{k}(d_{k}^{p})\) and the predict reduction \(Pd_{k}(d_{k}^{p})\), their radio is defined by
Therefore, the tensor trustregion model algorithm for solve (1.1) is stated as follows.
Algorithm 1
 Initial: :

Constants ρ, \(c\in (0,1)\), \(p=0\), \(\epsilon >0\), \(x_{0}\in \Re^{n},\,m>0\), and \(B_{0}=H_{0}^{1}\in \Re^{n}\times \Re ^{n}\) is a symmetric and positive definite matrix. Let \(k:=0\);
 Step 1: :

Stop if \(\Vert S(x_{k}) \Vert <\epsilon \) holds;
 Step 2: :

Solve (2.3) with \(\triangle =\triangle_{k}\) to obtain \(d_{k}^{p}\);
 Step 3: :

Compute \(Ad_{k}(d_{k}^{p})\), \(Pd_{k}(d_{k}^{p})\), and the radio \(r_{k}^{p}\). If \(r_{k}^{p}<\rho \), let \(p=p+1\), go to Step 2. If \(r_{k}^{p}\geq \rho \), go to the next step;
 Step 4: :

Set \(x_{k+1}=x_{k}+d_{k}^{p}\), \(y_{k}=S(x_{k+1})S(x _{k})\), update \(B_{k+1}=H_{k+1}^{1}\) by (2.4) if \(y_{k}^{T}d _{k}^{p}>0\), otherwise set \(B_{k+1}=B_{k}\);
 Step 5: :

Let \(k:=k+1\) and \(p=0\). Go to Step 1.
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
$$ \varOmega =\bigl\{ x\mid\beta (x)\leq \beta (x_{0})\bigr\} $$(3.1)is bounded.
 (B) :

On an open convex set \(\varOmega_{1}\) containing Ω, the nonlinear system \(S(x)\) is twice continuously differentiable.
 (C) :

The approximation relation
$$ \bigl\Vert \bigl[\nabla S(x_{k})B_{k} \bigr]S(x_{k}) \bigr\Vert =O\bigl(\bigl\Vert d_{k}^{p} \bigr\Vert \bigr) $$(3.2)is true, where \(d_{k}^{p}\) is the solution of the model (2.3).
 (D) :

On \(\varOmega_{1}\), the sequence matrices \(\{B_{k}\}\) are uniformly bounded, namely there exist constants \(0< M_{0}\leq M\) satisfying
$$ M_{s}\leq \Vert B_{k} \Vert \le M_{l} \quad \forall k. $$(3.3)
Assumption i (B) means that there exists a constant \(M_{L}>0\) satisfying
Based on the above assumptions and the definition of the model (2.3), we have the following lemma.
Lemma 3.1
Let \(d_{k}^{p}\) be the solution of (2.3), then the inequality
holds.
Proof
By the definition of \(d_{k}^{p}\) of (2.3), then, for any \(\alpha \in [0,1]\), we get
Therefore, we have
The proof is complete. □
Lemma 3.2
Let \(d_{k}^{p}\) be the solution of (2.3). Suppose that Assumption i holds and \(\{x_{k}\}\) is generated by Algorithm 1. Then we have
Proof
Using Assumption i, the definition of (2.5) and (2.6), we obtain
This completes the proof. □
Lemma 3.3
Let the conditions of Lemma 3.2 hold. We conclude that Algorithm 1 does not infinitely circle in the inner cycle (“Step 2–Step 3–Step 2”).
Proof
This lemma will be proved by contradiction. Suppose, at \(x_{k}\), that Algorithm 1 infinitely circles in the inner cycle, namely, \(r_{k}^{p}<\rho \) and \(c^{p}\rightarrow 0\) with \(p\rightarrow \infty \). This implies that \(\Vert g_{k} \Vert \geq \epsilon \), or the algorithm stops. Thus we conclude that \(\Vert d_{k}^{p} \Vert \leq \triangle_{k}=c^{p}\Vert g_{k} \Vert \rightarrow 0\) 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 \(r_{k}^{p}<\rho \). The proof is complete. □
Lemma 3.4
Suppose that the conditions of Lemma 3.3 holds. Then we conclude that \(\{x_{k}\}\subset \varOmega \) is true and \(\{\beta (x_{k})\}\) converges.
Proof
By the results of the above lemma, we get
Combining with Lemma 3.1 generates
Then \(\{x_{k}\}\subset \varOmega \) holds. By the case \(\beta (x_{k}) \geq 0\), we deduce that \(\{\beta (x_{k})\}\) converges. This completes its proof. □
Theorem 3.5
Suppose that the conditions of Lemma 3.3 hold and \(\{x_{k}\}\) is generated by Algorithm 1. Then Algorithm 1 either finitely stops or generates an infinite sequence \(\{x_{k}\}\) satisfying
Proof
Suppose that Algorithm 1 does not finitely stop. We need to obtain (3.8). Assume that
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
Let \(K=\{k\mid \Vert B_{k}S(x_{k}) \Vert \geq \varepsilon \}\) be an index set. Using Assumption i, the case \(\Vert B_{k}S(x_{k}) \Vert \geq \varepsilon\) (\(k\in K\)), and \(\Vert S(x_{k}) \Vert \) (\(k\in K\)) 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 \(\{\beta (x_{k})\}\) is convergent, thus
Then \(p_{k} \rightarrow +\infty \) when \(k\rightarrow +\infty \) and \(k\in K\). Therefore, for all \(k\in K\), it is reasonable for us to assume \(p_{k}\geq 1\). In the inner circle, by the determination of \(p_{k}\) (\(k\in K\)), let \(d_{k}'\) corresponding to the subproblem
be unacceptable. Setting \(x_{k+1}'=x_{k}+d_{k}'\) one has
Using Lemma 3.1 and the definition \(\triangle_{k}\) one has
Using Lemma 3.2 one gets
Thus, we obtain
Using \(p_{k}\rightarrow +\infty \) when \(k\rightarrow +\infty \) and \(k\in K\), 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 1
Trigonometric function
Initial guess: \(x_{0}=(\frac{101}{100n},\frac{101}{100n},\ldots , \frac{101}{100n})^{T}\).
Problem 2
Logarithmic function
Initial points: \(x_{0}=(1,1,\ldots ,1)^{T}\).
Problem 3
Broyden tridiagonal function ([7], pp. 471–472)
Initial points: \(x_{0}=(1,1,\ldots ,1)^{T}\).
Problem 4
Trigexp function ([7], p. 473)
Initial guess: \(x_{0}=(0,0,\ldots ,0)^{T}\).
Problem 5
Strictly convex function 1 ([18], p. 29). \(S(x)\) is the gradient of \(h(x)=\sum_{i=1}^{n}(e^{x_{i}}x_{i})\). We have
Initial points: \(x_{0}=(\frac{1}{n},\frac{2}{n},\ldots ,1)^{T}\).
Problem 6
Strictly convex function 2 ([18], p. 30). \(S(x)\) is the gradient of \(h(x)=\sum_{i=1}^{n}\frac{i}{10} (e^{x_{i}}x _{i})\). We have
Initial guess: \(x_{0}=(1,1,\ldots ,1)^{T}\).
Problem 7
Penalty function
Initial guess: \(x_{0}=(\frac{1}{3},\frac{1}{3},\ldots ,\frac{1}{3})^{T}\).
Problem 8
Variable dimensioned function
Initial guess: \(x_{0}=(1\frac{1}{n},1\frac{2}{n},\ldots ,0)^{T}\).
Problem 9
Discrete boundary value problem [15]
Initial points: \(x_{0}=(h(h1),h(2h1),\ldots ,h(nh1))\).
Problem 10
The discretized twopoint boundary value problem similar to the problem in [17]
with A is the \(n\times n\) tridiagonal matrix given by
and \(F(x)=(F_{1}(x),F_{2}(x),\ldots,F_{n}(x))^{T}\) with \(F_{i}(x)=\sin x _{i} 1\), \(i=1,2,\ldots,n\), and \(x=(50,0,50,0, \ldots )\).
Parameters: \(\rho =0.05\), \(\epsilon =10^{4}\), \(c=0.5\), \(p=3\), \(m=6\), \(H_{0}\) is the unit matrix.
The method for ( 1.3 ) and ( 2.3 ): the \(Dogleg\) method [22].
Codes experiments: run on a PC with an Intel Pentium(R) Xeon(R) E5507 CPU @2.27 GHz, 6.00 GB of RAM, and the Windows 7 operating system.
Codes software: MATLAB r2017a.
Stop rules: the program stops if \(\Vert S(x) \Vert \leq 1e{}4\) holds.
Other cases: we will stop the program if the iteration number is larger than a thousand.
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 CPUtime 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 trustregion 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 trustregion model is established and discussed.

(2)
the low workload update is used in this tensor trustregion model. In the future, we think this tensor trustregion model shall be more significant.
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Acknowledgements
The authors would like to thank the above the support funding. The authors also thank the referees and the editor for their valuable suggestions which greatly improve our paper.
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Songhua Wang and Shulun Liu are cofirst authors.
Funding
This work was supported by the National Natural Science Fund of China (Grant No. 11661009).
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Dr. Songhua Wang mainly analyzed the theory results and Shulun Liu has done the numerical experiments. All authors read and approved the final manuscript.
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Wang, S., Liu, S. A tensor trustregion model for nonlinear system. J Inequal Appl 2018, 343 (2018). https://doi.org/10.1186/s1366001819350
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DOI: https://doi.org/10.1186/s1366001819350
MSC
 65K05
 90C26
Keywords
 Tensor model
 Trust region
 Nonlinear equations
 BFGS formula
 Convergence