# A fast algorithm for globally solving Tikhonov regularized total least squares problem

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## Abstract

The total least squares problem with the general Tikhonov regularization can be reformulated as a one-dimensional parametric minimization problem (PM), where each parameterized function evaluation corresponds to solving an *n*-dimensional trust region subproblem. Under a mild assumption, the parametric function is differentiable and then an efficient bisection method has been proposed for solving (PM) in literature. In the first part of this paper, we show that the bisection algorithm can be greatly improved by reducing the initially estimated interval covering the optimal parameter. It is observed that the bisection method cannot guarantee to find the globally optimal solution since the nonconvex (PM) could have a local non-global minimizer. The main contribution of this paper is to propose an efficient branch-and-bound algorithm for globally solving (PM), based on a new underestimation of the parametric function over any given interval using only the information of the parametric function evaluations at the two endpoints. We can show that the new algorithm (BTD Algorithm) returns a global \(\epsilon \)-approximation solution in a computational effort of at most \(O(n^3/\sqrt{\epsilon })\) under the same assumption as in the bisection method. The numerical results demonstrate that our new global optimization algorithm performs even much faster than the improved version of the bisection heuristic algorithm.

## Keywords

Total least squares Tikhonov regularization Trust region subproblem Fractional program Lower bound Branch and bound## Mathematics Subject Classification

65F20 90C26 90C32 90C20## 1 Introduction

*E*and

*r*are the perturbations. For more details, we refer to [10, 23, 24] and references therein. Let \((E^*,r^*,x^*)\) be an optimal solution to the above minimization problem (1). It can be verified that \(E^*\) and \(r^*\) have a closed-form expression in terms of \(x^*\) as the problem (1) is a linear-equality constrained convex quadratic program with respect to

*E*and

*r*. Therefore, by eliminating

*E*and

*r*from (1), we obtain the following equivalent quadratic fractional program:

*A*

*b*] if it has a full column rank, see [9, 24].

*E*and

*r*, we can recast (3) as the following optimization problem with respect to

*x*[1, 14]:

*L*. Throughout this paper, we make the following assumption, which was firstly presented in [1]:

Let \(x^*\) be a globally optimal solution to (P). Then \(\alpha ^*=\Vert x^*\Vert ^2+1\) is an optimal solution of \((\mathrm PM)\). Algorithm TRTLSG starts from an initial interval covering \(\alpha ^*\), denoted by \([\alpha _{\min },\alpha _{\max }]\). As in [1], \(\alpha _{\min }\) is trivially set as \(1\,+\,\epsilon _1\), where \(\epsilon _1>0\) is a tolerance parameter. \(\alpha _{\max }\) is chosen in a closed form based on a tedious derivation of the upper bound of \(\Vert x^*\Vert \) under Assumption (4). Notice that the computational cost of Algorithm TRTLSG is proportional to \(\log (\alpha _{\max }-\,\alpha _{\min })\), where \(\alpha _{\max }-\alpha _{\min }\) is the length of the initial interval. Thus, in the first part of this paper, we try to improve the lower and upper estimations of \(\alpha ^*\). More precisely, we firstly establish a new closed-form upper bound of \(\alpha ^*\), which greatly improves the quality of the current estimation at the same computational cost. Secondly, a new lower bound of \(\alpha ^*\) is derived in place of the trivial setting \(\alpha _{\min }=1\,+\,\epsilon _1\). With the new setting of \(\alpha _{\min }\) and \(\alpha _{\max }\), the efficiency of Algorithm TRTLSG is greatly improved at least for all our tested numerical results.

The main contribution of this paper is to propose a two-layer dual approach for underestimating \(\mathcal {G}(\alpha )\) over any given interval, without additional computational cost except for evaluating \(\mathcal {G}(\alpha )\) at the two endpoints of the interval. With this high-quality underestimation, we develop an efficient branch-and-bound algorithm to solve the one-dimensional parametric reformulation \((\mathrm PM)\) (5). Our new algorithm guarantees to find a global \(\epsilon \)-approximation solution of \((\mathrm PM)\) in at most \(O(1/\epsilon )\) iterations and the computational effort in each iteration is \(O(n^3\log (1/\epsilon ))\). Under the additional assumption to make \(\mathcal {G}(\alpha )\) be differentiable, the number of iterations can be further reduced to \(O(1/\sqrt{\epsilon })\). Numerical results demonstrate that, in most cases, our new global optimization algorithm is much faster than the improved version of the heuristic Algorithm TRTLSG.

The remainder of the paper is organized as follows. In Sect. 2, we present some preliminaries and the bisection heuristic Algorithm TRTLSG. In Sect. 3, we establish new lower and upper bounds on the norm of any optimal solution of (P), with which the computation cost of Algorithm TRTLSG greatly decreases. In Sect. 4, we propose a new underestimation and then use it to develop an efficient branch-and-bound algorithm. The worst-case computational complexity is also analyzed. Numerical comparisons among the above two algorithms are reported in Sect. 5. Concluding remarks are made in Sect. 6.

Throughout the paper, the notation “:=” denotes “define”. \(v(\cdot )\) denotes the optimal objective value of the problem \((\cdot )\). *I* is the identity matrix. The matrix \(A\succ (\succeq )0\) stands for that *A* is positive (semi-)definite. The inner product of two matrices *A* and *B* are tr(\(AB^T\)). \(\mathrm{Range}(A)=\{Ax: x\in \mathbb {R}^n\}\) is the range space of *A*. The one-dimensional intervals \(\{x: a< x < b\}\) and \(\{x: a \le x \le b\}\) are denoted by (*a*, *b*) and [*a*, *b*], respectively. \(\lceil (\cdot )\rceil \) is the smallest integer larger than or equal to \((\cdot )\).

## 2 The bisection algorithm

In this section, we present the bisection algorithm, denoted by Algorithm TRTLSG in [1]. To begin with, we firstly list some preliminary results of (P) and \(\mathcal {G}(\alpha )\) defined in (5).

### Theorem 1

### Theorem 2

### Theorem 3

### Corollary 1

Theorem 3 supported many algorithms for solving (TRS), see, for example, [4, 6, 16, 18, 19, 21]. In this paper, for the tested medium-scale problems, we apply the solution approach based on the complete spectral decomposition [7].

### Theorem 4

([1]) \(\mathcal {G}(\alpha )\) is continuous over \([1,+\infty )\).

### Theorem 5

If the function \(\mathcal {G}(\alpha )\) is unimodal, Algorithm TRTLSG converges to the global minimizer of (P). It is proved to be true when \(L=I\) [1]. In general, this is not true. A counterexample (with \(m = n =4\), \(k = 3\)) is plotted in [1] to show that \(\mathcal {G}(\alpha )\) is not always unimodal. Thus, Algorithm TRTLSG could return a local non-global minimizer of (P).

## 3 Bounds on the norm of any globally optimal solution

In this section, we establish new lower and upper bounds on the norm of any globally optimal solution of (P). They help to greatly improve the efficiency of Algorithm TRTLSG.

### 3.1 A new lower bound

To our best knowledge, there is no nontrivial lower bound on the norm of any globally optimal solution of (P) except for the trivial setting \(1+\epsilon _1\) in [1]. In this subsection, in order to derive such a new lower bound, we firstly need a technical lemma.

### Lemma 1

### Proof

### Theorem 6

### Proof

*J*(

*x*) has a unique minimizer \(x^*=(A^{T}A+ \rho L^{T}L)^{-1}A^{T}b\). Since \(A^{T}b \ne 0\), we have \(x^*\ne 0\) and thus it holds that

Finally, we assume \(A^{T}b = 0\) and \(b\ne 0\). The relation between the initial setting of \(\alpha _{\min }\) and the quality of the approximation minimizer of \(\mathcal {G}(\alpha )\) over \(\{1\}\cup [\alpha _{\min },\alpha _{\max }]\) is established as follows.

### Proposition 1

### Proof

### 3.2 New upper bounds

In this subsection, we propose two improved upper bounds on the norm of any optimal solution of (P), one of which has the same computational cost as the upper bound given in Theorem 2.

*v*(SDP) gives a new upper bound of \(\Vert x^*\Vert ^2\). If strong duality holds for (22), then the new bound

*v*(SDP) is always not weaker than (7). But the computation of (SDP) is much more time-consuming than that of (7).

In the following, we propose a new upper bound of \(\Vert x^*\Vert ^2\) with the same computational effort as (7). The basic idea is directly following the original inequality (6) rather than (20)-(21).

### Theorem 7

### Proof

*v*(SDP) and (23), we do numerical experiments using the noise-free data of the first example presented in Sect. 5. The dimension

*n*varies from 20 to 3000 and the regularization parameter \(\rho \) is simply fixed at 0.5. The computational environment is presented in Sect. 5. We report the numerical results in Table 1. It can be seen that

*v*(SDP) gives the tightest upper bound with the highest computation cost. For each test instance, the new upper bound (23) is much tighter than the existing upper bound (7) in the same computational time. We can see that, for the instance of dimension 1000, Algorithm TRTLSG will save \(\log _2\left( \frac{1.97\times 10^{12}}{4.79\times 10^6}\right) \approx 15\) iterations if the new upper bound (23) is used to replace (7). From Columns 2–3 of Table 1, it is observed that the new lower bound (17), solved at a low computational cost, is much tighter than the trivial bound \(1+\epsilon _1=1.1\). For the instance of dimension 1000, replacing the trivial bound \(1+\epsilon _1\) with the new lower bound (17) will help Algorithm TRTLSG to save \(\log _2\left( \frac{1.64\times 10^{2}}{1.1}\right) \approx 7\) iterations.

n | New b.d. (17) | New b.d. (23) |
| |||||
---|---|---|---|---|---|---|---|---|

Time | \(\alpha _{\min }\) | Time | \(\alpha _{\max }\) | Time | \(\alpha _{\max }\) | Time | \(\alpha _{\max }\) | |

20 | 0.00 | 4.28 | 0.00 | 3.02 | 0.00 | 2.28 | 1.39 | 4.76 |

50 | 0.00 | 9.18 | 0.00 | 1.35 | 0.00 | 1.32 | 0.47 | 1.60 |

100 | 0.00 | 1.73 | 0.00 | 3.08 | 0.00 | 5.08 | 0.63 | 4.27 |

200 | 0.00 | 3.37 | 0.00 | 7.98 | 0.02 | 1.98 | 1.30 | 1.20 |

500 | 0.03 | 8.27 | 0.02 | 6.62 | 0.03 | 1.21 | 6.97 | 5.17 |

1000 | 0.09 | 1.64 | 0.09 | 1.97 | 0.09 | 4.79 | 25.75 | 1.66 |

1200 | 0.14 | 1.97 | 0.13 | 4.83 | 0.14 | 6.88 | 59.03 | 2.28 |

1500 | 0.22 | 2.46 | 0.20 | 1.45 | 0.20 | 1.07 | 102.81 | 3.37 |

1800 | 0.31 | 2.95 | 0.33 | 3.56 | 0.33 | 1.54 | 149.72 | 4.66 |

2000 | 0.39 | 3.28 | 0.41 | 6.00 | 0.41 | 1.90 | 199.24 | 5.63 |

2500 | 0.90 | 4.10 | 0.98 | 1.81 | 1.02 | 2.96 | 307.04 | 8.42 |

3000 | 2.42 | 4.92 | 2.44 | 4.46 | 2.36 | 4.26 | 470.16 | 1.17 |

## 4 Branch-and-bound algorithm based on a two-layer dual approach

In this section we firstly present a two-layer dual approach for underestimating \(\mathcal {G}(\alpha )\) (5) and then use it to develop an efficient branch-and-bound algorithm (Algorithm BB-TLD). The worst-case computational complexity is also analyzed.

### 4.1 A two-layer dual underestimation approach

The efficiency to solve (P) via (5) relies on an easy-to-compute and high-quality lower bound of \(\mathcal {G}(\alpha )\) (5) over any given interval \([\alpha _i,\alpha _{i+1}]\). The difficulty is that there seems to be no closed-form expression of \(\mathcal {G}(\alpha )\). In this subsection, we present a new approach for underestimating \(\mathcal {G}(\alpha )\).

### Theorem 8

### 4.2 A new branch-and-bound algorithm (Algorithm BB-TLD)

We show the details of applying Algorithm BB-TLD to solve the following example where \(\mathcal {G}(\alpha )\) is not unimodal with the setting \(\epsilon =10^{-6}\).

### Example 1

In order to study the worst-case computational complexity of our new algorithm, we need the following lemma.

### Lemma 2

### Proof

### Theorem 9

*U*is defined in (52), \(\alpha _{\min }>1\) and \(\alpha _{\max }\) are constant real numbers defined in Theorems 6 and 7, respectively. Moreover, suppose the assumption (15) holds for all \(\alpha >1\), in order to find a global \(\epsilon \)-approximation solution of \((\mathrm{P_G})\), our new algorithm requires at most

### Proof

Suppose \((LB,\alpha _i,\alpha _{i+1})\in T\) is selected to subdivide in the current iteration of our new algorithm. Then, we have \(LB=LB^*\). Without loss of generality, we assume that (50) holds in the interval \([\alpha _i,\alpha _{i+1}]\), since otherwise, it follows from Theorem 8 that \(LB=UB\) and hence the algorithm has to stop.

### Corollary 2

### Proof

## 5 Numerical experiments

We numerically test two examples. The first one is taken from Hansen’s Regularization Tools [11], where the function *shaw* is used to generate the matrix \(A_{\mathrm{true}}\in \mathbb {R}^{n\times n}\), the vector \(b_{\mathrm{true}}\in \mathbb {R}^n\) and the true solution \(x_{\mathrm{true}}\in \mathbb {R}^n\), i.e., we have \(A_{\mathrm{true}}x_\mathrm{true}=b_{\mathrm{ture}}\). Then, we add the white noise of level \(\sigma =0.05\), i.e., \(A=A_{\mathrm{true}}+\sigma E\), \(b=b_\mathrm{true}+\sigma e\), where *E* and *e* are generated from a standard normal distribution. In our experiments, the dimension *n* varies from 20 to 5000.

*blur*(

*n*, 3), which is taken from [11]. The true solution \(x_{\mathrm{true}}\in \mathbb {R}^{n}\) is obtained by stacking the columns of \(X\in \mathbb {R}^{32\times 32}\) one underneath the other and then normalizing it so that \(\Vert x_{\mathrm{true}}\Vert =1\), where \(X\in \mathbb {R}^{32\times 32}\) is the following two dimensional image:

*E*and

*e*are generated from a standard normal distribution. In our experiments, we let the level of the noise \(\sigma \) vary in \(\{0.01,0.03,0.05, 0.08,0.1,0.3,0.5,0.8,1.3,1.5,1.8,2.0\}\).

For the regularization matrix of the first example, we take \(L=get\_l(n,1)\), which is given in [11]. For the second example, as in [2], we set the regularization matrix *L* as the discrete approximation of the Laplace operator, which is standard in image processing [13]. The regularization parameter \(\rho \) is selected by using the L-curve method [12]. It corresponds to the L-shaped corner of the norm \(\Vert Lx\Vert ^2\) versus the fractional residual \(\Vert Ax-b\Vert ^2/(\Vert x\Vert ^2+1)\) for a various number of regularization parameters.

All the experiments are carried out in MATLAB R2014a and run on a server with 2.6 GHz dual-core processor and 32 GB RAM. We set the tolerance parameter \(\epsilon =10^{-6}\) for all the two algorithms. For each setting of the dimension or the level of noise in the above two examples, we independently and randomly generate 10 instances and then run the two algorithms. We report in Tables 2 and 3 the average of the numerical results for the 10 times running, where the average computational time is recorded in seconds and the symbol ‘#iter’ denotes the average of the number of iterations, i.e., the number of evaluating (TRS).

The average of the numerical results for ten times solving the first example with different dimension *n*

n | Algorithm TRTLSG | Algorithm BB-TLD | ||
---|---|---|---|---|

# iter | Time (s) | # iter | Time (s) | |

20 | 16.0 | 0.02 | 17.0 | 0.02 |

50 | 18.3 | 0.03 | 15.5 | 0.03 |

100 | 18.7 | 0.09 | 15.5 | 0.08 |

200 | 18.9 | 0.26 | 16.5 | 0.25 |

500 | 20.0 | 2.81 | 16.8 | 2.64 |

1000 | 20.5 | 10.49 | 16.1 | 9.10 |

1200 | 20.4 | 15.19 | 15.6 | 12.69 |

1500 | 21.1 | 24.07 | 18.0 | 22.40 |

1800 | 21.2 | 35.97 | 17.8 | 33.67 |

2000 | 20.8 | 43.88 | 17.8 | 43.10 |

2500 | 20.7 | 72.51 | 17.5 | 68.84 |

3000 | 21.8 | 125.16 | 16.2 | 102.76 |

4000 | 20.2 | 255.86 | 14.0 | 202.95 |

5000 | 20.0 | 448.39 | 14.5 | 366.50 |

The average of the numerical results for ten times solving the second example with a fixed dimension \(n=1024\) and different level of noise \(\sigma \)

\(\sigma \) | Algorithm TRTLSG | Algorithm BB-TLD | ||
---|---|---|---|---|

# iter | Time (s) | # iter | Time (s) | |

0.01 | 17.2 | 9.46 | 14.4 | 8.60 |

0.03 | 21.9 | 12.33 | 16.6 | 10.11 |

0.05 | 16.2 | 8.91 | 17.0 | 10.52 |

0.08 | 18.0 | 10.04 | 17.0 | 10.68 |

0.1 | 19.8 | 11.25 | 18.4 | 11.56 |

0.3 | 29.4 | 17.65 | 17.0 | 10.94 |

0.5 | 30.8 | 18.59 | 17.4 | 11.19 |

0.8 | 30.7 | 20.52 | 15.9 | 12.06 |

1.0 | 31.4 | 21.08 | 15.4 | 11.71 |

1.3 | 32.2 | 21.56 | 15.6 | 12.04 |

1.5 | 32.0 | 21.96 | 15.6 | 12.33 |

1.8 | 33.6 | 23.11 | 16.1 | 13.04 |

2.0 | 33.7 | 23.52 | 16.0 | 13.03 |

## 6 Conclusions

The total least squares problem with the general Tikhonov regularization (TRTLS) is a non-convex optimization problem with local non-global minimizers. It can be reformulated as a problem of minimizing the one-dimensional function \(\mathcal {G}(\alpha )\) over an interval, where \(\mathcal {G}(\alpha )\) is evaluated by solving an *n*-dimensional trust region subproblem. In literature, there is an efficient bisection-based heuristic algorithm for solving (TRTLS), denoted by Algorithm TRTLSG. It converges to the global optimal solution except for some exceptional examples with non-unimodal \(\mathcal {G}(\alpha )\). In this paper, we firstly improve the lower and upper bounds on the norm of the globally optimal solution. It helps to greatly improve the efficiency of Algorithm TRTLSG. For the global optimization of (TRTLS), we employ the adaptive branch-and-bound algorithm, based on a newly introduced two-layer dual approach for underestimating \(\mathcal {G}(\alpha )\) over any given interval. Our new algorithm (Algorithm BB-TLD) guarantees to find a global \(\epsilon \)-approximation solution in at most \(O(1/\epsilon )\) iterations and the computational effort in each iteration is \(O(n^3\log (1/\epsilon ))\). Under the same assumptions as in Algorithm TRTLSG, the number of iterations of our new algorithm can be further reduced to \(O(1/\sqrt{\epsilon })\). In our experiments, the practical iteration numbers are always less than twenty and seem to be independent of the dimension and the level of noise. Numerical results demonstrate that our global optimization algorithm is even faster than the improved version of Algorithm TRTLSG, which is a bisection-based heuristic algorithm. It is the future work to extend our two-layer dual underestimation approach to globally solve more structured non-convex optimization problems.

## Notes

### Acknowledgements

The authors are grateful to the two anonymous referees for their valuable comments and suggestions.

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