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Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations

  • * Corresponding author: Jinkui Liu

    * Corresponding author: Jinkui Liu 
supported by the National Natural Science Foundation of China (Grant number: 11571055), the fund of Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant number:KJ1501003) and Chongqing Three Gorges University(Grant number:14ZD-14).
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  • In this paper, we consider a multivariate spectral DY-type projection method for solving nonlinear monotone equations with convex constraints. The search direction of the proposed method combines those of the multivariate spectral gradient method and DY conjugate gradient method. With no need for the derivative information, the proposed method is very suitable to solve large-scale nonsmooth monotone equations. Under appropriate conditions, we prove the global convergence and R-linear convergence rate of the proposed method. The preliminary numerical results also indicate that the proposed method is robust and effective.

    Mathematics Subject Classification: Primary: 49M37, 65H10; Secondary: 65K05.

    Citation:

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  • Table 1.  The results of Problem 1 with given initial points

    Dim Initial points MSGP method Algorithm 2.1
    1000 X1 7/22/0.06/1.31135e-007 13/40/0.05/3.38669e-006
    X2 2/7/0.05/0.00000e+000 4/13/0.03/0.00000e+000
    X3 11/34/0.08/2.19964e-007 3/40/0.05/3.36081e-006
    X4 3/10/0.05/0.00000e+000 6/19/0.05/0.00000e+000
    X5 11/34/0.09/8.93938e-008 8/25/0.06/6.13420e-006
    3000 X1 2/7/0.05/0.00000e+000 13/40/0.06/5.81725e-006
    X2 11/34/0.25/4.90071e-006 4/13/0.03/0.00000e+000
    X3 3/10/0.06/0.00000e+000 13/40/0.08/5.80007e-006
    X4 11/34/0.30/2.69613e-006 6/19/0.05/0.00000e+000
    X5 7/22/0.31/1.31003e-007 10/31/0.42/2.65195e-006
    5000 X1 2/7/0.05/0.00000e+000 13/40/0.08/7.49753e-006
    X2 11/34/0.53/7.57273e-006 4/13/0.05/ 0.00000e+000
    X3 3/10/0.13/0.00000e+000 13/40/0.09/7.48340e-006
    X4 12/37/0.78/8.40744e-008 6/19/0.05/0.00000e+000
    X5 7/22/0.64/1.30968e-007 10/31/1.05/3.78182e-006
    10000 X1 2/7/0.06/0.00000e+000 14/43/0.14/2.89200e-006
    X2 11/34/1.77/6.56131e-006 4/13/0.06/0.00000e+000
    X3 3/10/0.36/0.00000e+000 14/43/0.16/2.88949e-006
    X4 13/40/3.00/6.90437e-008 6/19/0.08/0.00000e+000
    X5 7/22/2.23/1.30940e-007 10/31/3.20/5.11269e-006
    12000 X1 2/7/0.06/0.00000e+000 14/43/0.17/3.16733e-006
    X2 11/34/2.48/6.34881e-006 4/13/0.06/0.00000e+000
    X3 3/10/0.47/0.00000e+000 14/43/0.16/3.16500e-006
    X4 13/40/4.53/6.04693e-008 6/19/0.09/0.00000e+000
    X5 7/22/3.13/1.30935e-007 10/31/4.67/5.34936e-006
     | Show Table
    DownLoad: CSV

    Table 2.  The results of Problem 2 with given initial points

    Dim Initial points MSGP method Algorithm 2.1
    1000 X1 290/1164/1.28/9.81195e-006 33/161/0.05/9.43667e-006
    X2 285/1153/1.19/8.96144e-006 56/294/0.06/9.59066e-006
    X3 86/381/0.17/9.29618e-006 37/184/0.05/8.83590e-006
    X4 61/275/0.30/9.56339e-006 62/330/0.06/9.97633e-006
    X5 361/1457/1.55/8.71650e-006 47/246/0.23/8.93449e-006
    3000 X1 61/281/0.98/9.44368e-006 44/219/0.09/8.60313e-006
    X2 347/1390/9.69/9.33905e-006 51/289/0.11/9.89992e-006
    X3 110/467/2.80/8.61958e-006 38/189/0.08/7.07861e-006
    X4 65/295/0.89/9.73034e-006 47/275/0.11/6.69019e-006
    X5 361/1457/10.36/8.71650e-006 47/246/1.31/8.93449e-006
    5000 X1 73/324/3.75/9.96592e-006 57/291/0.19/9.69435e-006
    X2 305/1224/22.45/9.99741e-006 61/326/0.19/9.23331e-006
    X3 99/420/6.53/9.97198e-006 44/220/0.14/8.90390e-006
    X4 64/285/4.08/8.54432e-006 54/292/0.19/9.52366e-006
    X5 361/1457/26.92/8.71650e-006 47/246/3.28/8.93449e-006
    10000 X1 63/280/13.44/8.48262e-006 43/212/0.25/6.39422e-006
    X2 367/1471/101.08/9.69111e-006 48/249/0.52/9.16252e-006
    X3 79/353/18.42/8.41347e-006 65/331/0.63/8.97609e-006
    X4 120/498/16.27/8.51405e-006 64/357/0.41/9.82673e-006
    X5 361/1457/101.25/8.71650e-006 47/246/12.08/8.93449e-006
    12000 X1 67/301/15.72/9.73091e-006 43/211/0.30/6.68736e-006
    X2 390/1562/154.20/8.01930e-006 66/467/0.52/8.76219e-006
    X3 82/362/25.55/9.73575e-006 56/292/0.39/9.27446e-006
    X4 134/548/49.52/9.58265e-006 57/311/0.44/8.42381e-006
    X5 361/1457/144.38/8.71650e-006 47/246/17.19/8.93449e-006
     | Show Table
    DownLoad: CSV

    Table 3.  The results of Problem 3 with given initial points

    Dim Initial points MSGP method Algorithm 2.1
    1000 X1 54/226/0.09/9.68368e-006 43/264/0.06/9.77486e-006
    X2 556/4532/1.11/9.59335e-006 36/230/0.05/7.01507e-006
    X3 1004/9105/1.70/9.64279e-006 48/311/0.06/8.53004e-006
    X4 871/7511/1.70/9.98355e-006 51/318/0.06/8.62714e-006
    X5 58/264/0.30/9.42953e-006 39/257/0.16/8.20033e-006
    3000 X1 53/213/0.16/7.47095e-006 53/335/0.14/7.17598e-006
    X2 628/5226/4.92/9.53246e-006 36/226/0.09/6.14965e-006
    X3 1237/12041/7.30/9.87899e-006 47/303/0.13/7.53500e-006
    X4 1059/9647/13.73/9.81351e-006 54/386/0.16/9.61675e-006
    X5 58/264/1.73/9.42953e-006 39/257/1.11/8.20033e-006
    5000 X1 51/212/0.20/8.70652e-006 49/311/0.19/7.89292e-006
    X2 635/5353/10.84/9.92649e-006 39/247/0.16/6.13645e-006
    X3 1397/13868/16.84/9.98769e-006 77/531/0.30/7.77119e-006
    X4 1093/10060/36.78/9.97031e-006 84/687/0.36/5.17010e-006
    X5 58/264/4.27/9.42953e-006 39/257/2.75/8.20033e-006
    10000 X1 61/243/0.45/8.97371e-006 56/365/0.41/8.54468e-006
    X2 206/1273/16.94/9.54721e-006 42/275/0.30/5.75051e-006
    X3 1739/18415/57.48/9.96405e-006 61/418/0.45/6.13008e-006
    X4 1102/10030/135.05/9.77795e-006 4/125/0.69/0.00000e+000
    X5 58/264/15.72/9.42953e-006 39/257/10.11/8.20033e-006
    12000 X1 65/267/0.58/9.69254e-006 52/337/0.45/8.80176e-006
    X2 655/5624/48.14/9.97926e-006 42/276/0.36/8.48727e-006
    X3 1439/14609/47.84/9.95483e-006 58/396/0.52/9.34846e-006
    X4 1037/9446/181.69/9.44940e-006 4/71/103.92/0.00000e+000
    X5 58/264/22.38/9.42953e-006 39/257/14.28/8.20033e-006
     | Show Table
    DownLoad: CSV

    Table 4.  The results of Problem 4 with given initial points

    Dim Initial points MSGP method Algorithm 2.1
    1000 X1 194/1015/0.59/9.66433e-006 30/184/0.05/5.70062e-006
    X2 117/619/0.20/9.09560e-006 58/375/0.08/8.92285e-006
    X3 157/833/0.24/9.74765e-006 75/484/0.08/6.43351e-006
    X4 159/838/0.25/9.55038e-006 70/453/0.06/6.92855e-006
    X5 156/800/0.52/9.96648e-006 55/354/0.06/8.13127e-006
    3000 X1 174/908/3.66/9.87344e-006 94/610/0.19/9.67519e-006
    X2 164/839/0.89/9.10283e-006 43/272/0.09/4.67179e-006
    X3 168/886/0.98/9.48568e-006 75/486/0.16/8.24368e-006
    X4 213/1147/1.39/9.27139e-006 61/394/0.13/5.88525e-006
    X5 183/990/3.84/9.99259e-006 62/396/0.13/8.46852e-006
    5000 X1 174/915/6.94/9.56987e-006 42/267/0.13/9.61170e-006
    X2 163/840/1.94/9.18817e-006 63/404/0.19/9.53629e-006
    X3 186/1014/2.59/9.83818e-006 67/431/0.20/9.62859e-006
    X4 211/1127/2.70/9.61085e-006 62/402/0.19/6.30592e-006
    X5 170/881/9.19/9.70906e-006 41/261/0.14/7.03502e-006
    10000 X1 181/969/30.69/9.49929e-006 29/178/0.17/9.26476e-006
    X2 161/808/7.27/8.99994e-006 31/193/0.19/8.57493e-006
    X3 182/983/7.89/9.62444e-006 78/504/0.94/8.99403e-006
    X4 206/1069/8.33/9.78939e-006 40/250/0.23/9.71324e-006
    X5 182/972/30.75/9.75983e-006 68/439/0.38/8.26113e-006
    12000 X1 190/1007/59.33/9.43919e-006 52/331/0.34/8.48201e-006
    X2 177/920/9.55/9.60849e-006 68/467/0.45/6.51248e-006
    X3 180/960/11.08/9.56932e-006 85/549/0.55/9.43656e-006
    X4 200/1059/11.67/9.87597e-006 28/171/0.19/8.94950e-006
    X5 174/936/34.61/9.79661e-006 70/453/0.47/7.91811e-006
     | Show Table
    DownLoad: CSV

    Table 5.  The results with initial points randomly generated from (0, 1)

    Dim MSGP method Algorithm 2.1
    Problem 1 1000 10/31/0.08/1.29174e-007 6/19/0.03/0.00000e+000
    10/31/0.05/3.60715e-006 6/19/0.00/0.00000e+000
    10/31/0.05/1.31904e-007 6/19/0.02/0.00000e+000
    3000 11/34/0.30/4.53840e-006 6/19/0.03/0.00000e+000
    11/34/0.27/6.94297e-006 6/19/0.02/0.00000e+000
    11/34/0.25/9.30472e-006 6/19/0.02/0.00000e+000
    5000 11/34/0.72/9.94443e-006 6/19/0.05/0.00000e+000
    12/37/0.75/1.10770e-006 6/19/0.03/0.00000e+000
    12/37/0.72/9.50443e-008 6/19/0.03/0.00000e+000
    1000013/40/3.03/8.74543e-0086/19/0.08/0.00000e+000
    12/37/2.67/6.56124e-006 6/19/0.06/0.00000e+000
    12/37/2.64/4.66009e-006 6/19/0.05/0.00000e+000
    12000 13/40/4.13/1.19951e-007 6/19/0.09/0.00000e+000
    14/43/4.67/1.13842e-007 6/19/0.06/0.00000e+000
    13/40/4.13/1.98039e-007 6/19/0.06/0.00000e+000
    Problem 2 1000 105/506/0.33/9.97726e-006 84/487/0.09/7.86183e-006
    104/482/0.28/9.49717e-006 81/465/0.06/8.31377e-006
    >113/554/0.31/9.22890e-006 83/449/0.06/9.84927e-006
    3000 126/650/1.88/9.29974e-006 91/515/0.22/9.63881e-006
    139/677/2.05/8.98283e-006 94/521/0.19/8.70734e-006
    133/639/1.94/9.51118e-006 98/532/0.22/7.45756e-006
    5000 143/727/5.03/8.99648e-006 97/565/0.39/9.72677e-006
    134/692/5.02/7.94479e-006 100/567/0.38/8.99827e-006
    141/695/5.00/8.93255e-006 102/650/0.41/9.83925e-006
    10000 154/834/20.17/9.55153e-006 93/545/0.70/7.86166e-006
    152/796/20.00/8.59073e-006 122/745/0.92/8.76303e-006
    154/794/20.20/9.43874e-006 127/863/0.98/9.20702e-006
    12000 162/855/30.14/9.10526e-006 122/748/1.17/9.67756e-006
    163/854/30.45/9.97329e-006 114/805/1.16/8.33603e-006
    158/828/29.94/9.08543e-006 112/751/1.08/9.98871e-006
     | Show Table
    DownLoad: CSV

    Table 6.  The results with initial points randomly generated from (0, 1)

    Dim MSGP method Algorithm 2.1
    Problem 3 1000 248/1765/0.52/8.14403e-006 66/443/0.09/5.73224e-006
    233/1659/0.44/8.60251e-006 73/505/0.08/8.37567e-006
    265/1839/0.53/8.07376e-006 61/410/0.06/6.30228e-006
    3000 291/2170/2.63/6.57364e-006 77/535/0.23/9.31672e-006
    284/2163/2.41/9.63384e-006 88/612/0.23/9.82561e-006
    287/2196/2.55/9.97633e-006 85/583/0.23/7.58095e-006
    5000 293/2317/6.02/9.41072e-006 106/758/0.58/8.60162e-006
    300/2320/6.19/9.59322e-006 89/635/0.44/8.10417e-006
    286/2259/5.86/7.50859e-006 108/784/0.56/6.29653e-006
    10000 262/2147/17.13/9.37057e-006 181/1544/2.89/7.61750e-006
    296/2454/18.55/8.82342e-006 94/645/1.03/9.49197e-006
    277/2187/18.78/8.03849e-006 65/435/0.72/5.90105e-006
    12000 378/3238/32.92/6.12712e-006 82/601/1.02/9.99908e-006
    301/2446/28.58/9.40122e-006 139/1096/2.63/6.13477e-006
    300/2497/26.70/9.76692e-006 148/1137/2.44/8.01365e-006
    Problem 4 1000 255/1646/0.55/9.88522e-006 148/969/0.14/8.52252e-006
    315/2190/0.64/8.58459e-006 167/1132/0.14/7.14294e-006
    263/1779/0.51/9.78341e-006 152/998/0.11/6.83836e-006
    3000 266/1864/2.59/9.68351e-006 204/1466/0.52/8.15543e-006
    269/1792/2.56/9.50426e-006 176/1151/0.33/7.26404e-006
    303/2094/2.95/7.70559e-006 180/1257/0.42/7.24683e-006
    5000 322/2299/7.41/6.83957e-006 208/1469/0.91/5.80054e-006
    265/1790/5.94/9.84065e-006 181/1202/0.64/9.43111e-006
    260/2023/5.14/9.80395e-006 195/1315/0.77/6.31066e-006
    10000 271/2024/20.59/8.91775e-006 217/1519/2.16/6.92039e-006
    283/1997/21.91/9.98757e-006 216/1535/2.09/9.27616e-006
    272/2002/19.44/8.12148e-006 198/1299/1.64/7.45635e-006
    12000 218/1786/13.67/5.52117e-006 235/1728/3.33/7.46955e-006
    309/2230/32.14/9.85930e-006 231/1779/3.78/8.69710e-006
    248/1926/24.22/8.80377e-006 209/1463/2.95/8.14360e-006
     | Show Table
    DownLoad: CSV
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