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A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems

  • * Corresponding author: Chen Ling

    * Corresponding author: Chen Ling 

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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  • 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.

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

    Citation:

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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

    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

    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

    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
    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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