A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems

Wanbin Tong; Hongjin He; Chen Ling; Liqun Qi

doi:10.3934/naco.2020042

Article Contents

2020, Volume 10, Issue 4: 425-437. Doi: 10.3934/naco.2020042

This issue Previous Article Preface Next Article On box-constrained total least squares problem

A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems

Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

^* Corresponding author: Chen Ling
^* Corresponding author: Chen Ling

Received: March 2020

Revised: September 2020

Published: September 2020

H. He and C. Ling were supported in part by National Natural Science Foundation of China (Nos. 11771113 and 11971138) and Natural Science Foundation of Zhejiang Province (Nos. LY20A010018, LY19A010019, and LD19A010002)

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Abstract

The tensor eigenvalue complementarity problem (TEiCP) is a higher order extension model of the classical matrix eigenvalue complementarity problem (EiCP), which has been studied extensively in the literature from theoretical perspective to algorithmic design. Due to the high nonlinearity resulted by tensors, the corresponding TEiCPs are often not easy to be solved directly by the algorithms tailored for EiCPs. In this paper, we introduce a nonmonotone spectral projected gradient (NSPG) method equipped with a positive Barzilai-Borwein step size to find a solution of TEiCPs. A series of numerical experiments show that the proposed NSPG method can greatly improve the efficiency of solving TEiCPs in terms of taking much less computing time for higher dimensional cases. Moreover, computational results show that our NSPG method is less sensitive to choices of starting points than some state-of-the-art algorithms.

Keywords:

Tensor eigenvalue complementatrity problem,
Projected gradient method,
Barzilai-Borwein step size,
Polynomial optimization.

Mathematics Subject Classification: Primary: 15A68, 90C30; Secondary: 90C33.

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Figure 1. Mean and standard deviation of iterations and computing time of ten trials with random starting points for the case of computing Pareto $ H $-eigenpairs. From top to bottom corresponding to Example 4, 5, and 6, respectively

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Figure 2. Mean and standard deviation of iterations and computing time of ten trials with random starting points for the case of computing Pareto $ B $-eigenpairs. The first and second rows correspond to Example 7 and 8, respectively

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Figure 3. Mean and standard deviation of iterations and computing time of ten trials with random starting points for the case of computing Pareto $ Z $- and $ H $-eigenpairs, respectively. The first and second rows correspond to Example 9 and 10, respectively

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Algorithm 1: The nonmonotone SPG method for TEiCPs

1: Let $ \varepsilon>0 $ be a tolerance of termination, $ \sigma,\rho\in(0,1) $, $ M\ge1 $ be an integer, $ 0<\beta_{\rm min}<\beta_{\rm max} $, and $ x_0\ge0 $ be an initial point. Set $ \beta_0=1/||\varphi(x_0)|| $ and $ k=0 $.
2: Compute $ z_k=P_{{\Delta}^n}(x_k+\beta_k \varphi(x_k)) $, and the direction $ d_{\beta_k}(x_k)=z_k-x_k $.
3: If $ ||d_{\beta_k}(x_k)||\leq\varepsilon $ then stop, the output $ \lambda=\frac{{{\mathcal A}} x_k^m}{{{\mathcal B}} x_k^m} $ is a Pareto-eigenvalue and $ x_k $ is the corresponding Preto-eigenvector of TEiCP.
4: Set $ \alpha_k=\rho^i $, where $ i\ge 0 $ is the smallest integer, such that
$\begin{equation} \Phi(x_k+\alpha_k d_{\beta_k}(x_k))\ge \min_{0\le j\le {\min\{k, M-1\}}}\Phi(x_{k-j})+ \sigma\alpha_k \varphi(x_k)^\top d_{\beta_k}(x_k), \;\;\;\;(7)\end{equation}$
and let $ x_{k+1}=x_k+\alpha_k d_{\beta_k}(x_k) $.
5: Let $ s_k=x_{k+1}-x_k $ and $ y_k=\varphi(x_{k+1})-\varphi(x_{k}) $. Update
$\beta_{k+1} = \max \left\{\beta_{\min},\min\left\{\beta_{\max},\left|\frac{\langle s_k,s_k\rangle}{\langle s_k,y_k\rangle}\right|\right\}\right\}.$
Go to Step 2.

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Table 1. Computational results for Examples 1-10

Exam	(m, n)	NSPG(M=1)	NSPG(M=5)	NSPG(M=15)	SPG1	SPP
Exam	(m, n)	its / iits / time / err	its / iits / time / err	its / iits / time / err	its / iits / time / err	its / time / err
1	(4, 3)	9 / 3 / 0.06 / 2.4$ \times 10^{-7} $	9 / 3 / 0.05 / 2.4$ \times 10^{-7} $	9 / 3 / 0.05 / 2.4$ \times 10^{-7} $	31 / 78 / 0.28 / 5.6$ \times 10^{-7} $	19 / 0.08 / 9.0$ \times 10^{-7} $
2	(4, 5)	2 / 0 / 0.02 / 0	2 / 0 / 0.00 / 0	2 / 0 / 0.02 / 0	2 / 2 / 0.02 / 0	7 / 0.03 / 2.3$ \times 10^{-11} $
3	(4, 3)	8 / 1 / 0.03 / 1.0$ \times 10^{-6} $	8 / 0 / 0.03 / 1.1$ \times 10^{-8} $	8 / 0 / 0.03 / 1.1$ \times 10^{-8} $	11 / 26 / 0.09 / 2.9$ \times 10^{-7} $	5 / 0.02 / 3.9$ \times 10^{-7} $
	(4, 50)	35 / 22 / 1.08 / 7.0$ \times 10^{-7} $	35 / 2 / 0.69 / 5.7$ \times 10^{-7} $	32 / 1 / 0.63 / 5.8$ \times 10^{-7} $	83 / 172 / 3.19 / 9.8$ \times 10^{-7} $	147 / 2.78 / 9.9$ \times 10^{-7} $
4	(4, 75)	31 / 16 / 3.84 / 7.1$ \times 10^{-7} $	31 / 4 / 2.77 / 3.9$ \times 10^{-7} $	32 / 3 / 2.75 / 1.2$ \times 10^{-7} $	101 / 209 / 16.08 / 4.5$ \times 10^{-10} $	136 / 10.66 / 9.9$ \times 10^{-7} $
	(4,100)	34 / 30 / 15.75 / 5.9$ \times 10^{-7} $	32 / 17 / 11.78 / 5.9$ \times 10^{-7} $	32 / 17 / 12.14 / 5.9$ \times 10^{-7} $	87 / 180 / 42.50 / 9.5$ \times 10^{-7} $	134 / 548.48 / 9.4$ \times 10^{-7} $
	(4, 50)	19 / 6 / 0.50 / 8.0$ \times 10^{-7} $	21 / 0 / 0.41 / 3.4$ \times 10^{-7} $	21 / 0 / 0.41 / 3.4$ \times 10^{-7} $	21 / 31 / 0.59 / 4.2$ \times 10^{-7} $	250 / 4.64 / 9.9$ \times 10^{-7} $
5	(4, 75)	22 / 5 / 2.31 / 3.3$ \times 10^{-7} $	19 / 0 / 1.53 / 3.2$ \times 10^{-7} $	19 / 0 / 1.53 / 3.2$ \times 10^{-7} $	35 / 66 / 5.41 / 3.4$ \times 10^{-8} $	236 / 18.81 / 9.9$ \times 10^{-7} $
	(4,100)	21 / 8 / 7.58 / 2.2$ \times 10^{-7} $	24 / 1 / 6.47 / 6.9$ \times 10^{-7} $	25 / 0 / 6.11 / 7.1$ \times 10^{-7} $	29 / 56 / 13.47 / 5.1$ \times 10^{-7} $	323 / 1348.98 / 9.7$ \times 10^{-7} $
	(4, 50)	38 / 16 / 1.03 / 2.9$ \times 10^{-7} $	41 / 2 / 0.80 / 1.7$ \times 10^{-7} $	41 / 1 / 0.78 / 8.0$ \times 10^{-7} $	79 / 169 / 3.14 / 8.2$ \times 10^{-7} $	114 / 2.14 / 9.9$ \times 10^{-7} $
6	(4, 75)	58 / 43 / 7.94 / 6.2$ \times 10^{-7} $	48 / 16 / 5.20 / 6.7$ \times 10^{-7} $	50 / 1 / 4.31 / 8.9$ \times 10^{-7} $	186 / 385 / 29.63 / 9.7$ \times 10^{-7} $	369 / 28.98 / 1.0$ \times 10^{-6} $
	(4,100)	32 / 14 / 11.56 / 8.8$ \times 10^{-7} $	37 / 2 / 10.16 / 3.9$ \times 10^{-7} $	37 / 2 / 10.42 / 3.9$ \times 10^{-7} $	70 / 157 / 37.42 / 9.6$ \times 10^{-7} $	195 / 842.20 / 9.5$ \times 10^{-7} $
	(4, 50)	291 / 390 / 12.41 / 9.3$ \times 10^{-7} $	204 / 147 / 6.42 / 8.2$ \times 10^{-7} $	163 / 59 / 4.31 / 5.2$ \times 10^{-7} $	- / - / - / 1.7$ \times 10^{-5} $	- / - / 2.4$ \times 10^{-5} $
7	(4, 75)	156 / 176 / 27.72 / 9.0$ \times 10^{-7} $	178 / 113 / 24.14 / 5.3$ \times 10^{-7} $	155 / 30 / 15.41 / 3.5$ \times 10^{-7} $	567 / 1160 / 90.89 / 9.9$ \times 10^{-7} $	- / - / 4.0$ \times 10^{-5} $
	(4,100)	117 / 131 / 59.53 / 5.8$ \times 10^{-7} $	125 / 106 / 54.86 / 7.3$ \times 10^{-7} $	114 / 34 / 37.16 / 6.7$ \times 10^{-7} $	325 / 680 / 162.83 / 1.0$ \times 10^{-6} $	- / - / 4.3$ \times 10^{-5} $
	(4, 50)	21 / 20 / 0.75 / 1.7$ \times 10^{-8} $	21 / 20 / 0.75 / 1.7$ \times 10^{-8} $	21 / 20 / 0.77 / 1.7$ \times 10^{-8} $	645 / 650 / 11.78 / 1.0$ \times 10^{-6} $	- / - / 1.2$ \times 10^{-4} $
8	(4, 75)	22 / 23 / 3.50 / 7.8$ \times 10^{-8} $	34 / 23 / 4.41 / 2.0$ \times 10^{-8} $	38 / 22 / 4.63 / 1.2$ \times 10^{-10} $	738 / 741 / 59.89 / 9.9$ \times 10^{-7} $	- / - / 4.8$ \times 10^{-4} $
	(4,100)	26 / 27 / 14.19 / 3.4$ \times 10^{-9} $	26 / 23 / 13.02 / 3.4$ \times 10^{-9} $	26 / 23 / 12.50 / 3.4$ \times 10^{-9} $	916 / 942 / 236.67 / 1.0$ \times 10^{-6} $	- / - / 3.5$ \times 10^{-4} $
9	(4, 50)	43 / 25 / 1.28 / 1.4$ \times 10^{-7} $	38 / 3 / 0.77 / 2.9$ \times 10^{-7} $	35 / 2 / 0.72 / 3.7$ \times 10^{-7} $	390 / 787 / 14.39 / 9.3$ \times 10^{-7} $	194 / 3.61 / 9.7$ \times 10^{-7} $
Z-eig	(4, 75)	71 / 55 / 9.80 / 8.3$ \times 10^{-7} $	63 / 18 / 6.23 / 6.2$ \times 10^{-7} $	74 / 1 / 5.78 / 7.3$ \times 10^{-7} $	328 / 697 / 52.91 / 2.9$ \times 10^{-7} $	496 / 36.58 / 1.0$ \times 10^{-6} $
	(4,100)	238 / 297 / 131.00 / 9.4$ \times 10^{-7} $	120 / 61 / 42.97 / 5.1$ \times 10^{-7} $	138 / 27 / 39.31 / 8.3$ \times 10^{-7} $	231 / 470 / 112.47 / 6.1$ \times 10^{-7} $	295 / 1235.41 / 9.8$ \times 10^{-7} $
10	(3, 50)	55 / 31 / 0.33 / 9.7$ \times 10^{-7} $	50 / 7 / 0.22 / 1.0$ \times 10^{-6} $	58 / 0 / 0.22 / 1.2$ \times 10^{-7} $	133 / 284 / 1.08 / 3.8$ \times 10^{-7} $	121 / 0.48 / 9.7$ \times 10^{-7} $
H-eig	(3, 75)	61 / 34 / 0.38 / 4.0$ \times 10^{-7} $	45 / 6 / 0.20 / 4.8$ \times 10^{-8} $	45 / 2 / 0.20 / 1.0$ \times 10^{-7} $	288 / 593 / 2.39 / 1.9$ \times 10^{-8} $	158 / 0.72 / 9.9$ \times 10^{-7} $
	(3,100)	51 / 34 / 0.41 / 5.4$ \times 10^{-7} $	52 / 19 / 0.33 / 3.9$ \times 10^{-7} $	56 / 8 / 0.31 / 5.3$ \times 10^{-7} $	320 / 726 / 3.38 / 2.7$ \times 10^{-8} $	407 / 45.97 / 9.5$ \times 10^{-7} $
10	(4, 50)	106 / 84 / 3.47 / 1.4$ \times 10^{-7} $	64 / 16 / 1.45 / 6.2$ \times 10^{-7} $	53 / 4 / 1.08 / 9.9$ \times 10^{-7} $	330 / 768 / 14.19 / 1.7$ \times 10^{-7} $	173 / 3.45 / 9.5$ \times 10^{-7} $
H-eig	(4, 75)	82 / 57 / 10.81 / 1.1$ \times 10^{-7} $	64 / 10 / 5.70 / 6.8$ \times 10^{-7} $	68 / 1 / 5.31 / 5.8$ \times 10^{-7} $	576 / 1325 / 102.00 / 3.5$ \times 10^{-7} $	581 / 45.23 / 9.9$ \times 10^{-7} $
	(4,100)	70 / 40 / 27.61 / 8.7$ \times 10^{-7} $	114 / 57 / 40.80 / 5.1$ \times 10^{-7} $	97 / 4 / 25.73 / 3.5$ \times 10^{-7} $	218 / 518 / 124.77 / 7.1$ \times 10^{-7} $	372 / 1573.88 / 9.9$ \times 10^{-7} $
10	(5, 40)	32 / 17 / 15.00 / 5.9$ \times 10^{-7} $	36 / 4 / 12.47 / 6.1$ \times 10^{-7} $	35 / 2 / 11.08 / 5.8$ \times 10^{-7} $	615 / 1394 / 411.41 / 7.4$ \times 10^{-7} $	562 / 166.73 / 9.7$ \times 10^{-7} $
H-eig	(5, 50)	137 / 164 / 251.84 / 6.7$ \times 10^{-7} $	97 / 61 / 136.81 / 3.8$ \times 10^{-7} $	58 / 7 / 57.98 / 8.8$ \times 10^{-7} $	320 / 889 / 750.80 / 9.7$ \times 10^{-7} $	572 / 501.31 / 9.7$ \times 10^{-7} $
	(5, 60)	67 / 52 / 248.39 / 3.3$ \times 10^{-7} $	92 / 40 / 279.64 / 6.8$ \times 10^{-7} $	93 / 24 / 242.63 / 5.1$ \times 10^{-7} $	265 / 669 / 1365.61 / 7.6$ \times 10^{-7} $	347 / 724.52 / 9.9$ \times 10^{-7} $
The symbol '-' here means that the number of iterations exceeds 1000.

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