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Article Contents

# A diagonal PRP-type projection method for convex constrained nonlinear monotone equations

• * Corresponding author: Hassan Mohammad
• Iterative methods for nonlinear monotone equations do not require the differentiability assumption on the residual function. This special property of the methods makes them suitable for solving large-scale nonsmooth monotone equations. In this work, we present a diagonal Polak-Ribi$\grave{e}$re-Polyak (PRP) conjugate gradient-type method for solving large-scale nonlinear monotone equations with convex constraints. The search direction is a combine form of a multivariate (diagonal) spectral method and a modified PRP conjugate gradient method. Proper safeguards are devised to ensure positive definiteness of the diagonal matrix associated with the search direction. Based on Lipschitz continuity and monotonicity assumptions the method is shown to be globally convergent. Numerical results are presented by means of comparative experiments with recently proposed multivariate spectral Dai-Yuan-type (J. Ind. Manag. Optim. 13 (2017) 283-295) and Wei-Yao-Liu-type (Int. J. Comput. Math. 92 (2015) 2261-2272) conjugate gradient methods.

Mathematics Subject Classification: Primary: 49M37, 65H10, 65K05; Secondary: 90C56.

 Citation:

• Figure 1.  Performance profile with respect to number of iterations (ITER)

Figure 2.  Performance profile with respect to number of function evaluations

Figure 3.  Performance profile with respect to CPU time

Table 1.  The initial points used for the test problems

 INITIAL POINT VALUE $x_1$ $(1, 1, \ldots , 1)^T$ $x_2$ $(0.1, 0.1, \ldots , 0.1)^T$ $x_3$ $\bigl(\frac{1}{2}, \frac{1}{2^2}, \ldots , \frac{1}{2^n}\bigr)^T$ $x_4$ $\bigl(0, 1-\frac{1}{2}, \ldots , 1-\frac{1}{n}\bigr)^T$ $x_5$ $\bigl(0, \frac{1}{n}, \ldots , \frac{n-1}{n}\bigr)^T$ $x_6$ $\bigl(1, \frac{1}{2}, \ldots , \frac{1}{n}\bigr)^T$ $x_7$ $\bigl(n-\frac{1}{n}, n- \frac{2}{n}, \ldots , n-1 \bigr)^T$ $x_8$ $\bigl(\frac{1}{n}, \frac{2}{n}, \ldots , 1\bigr)^T$

Table 2.  Numerical Results for DPPM, MDYP and WYLP for Problem 1 with given initial points and dimension, $f$ represents failure

 DPPM MDYP WYLP DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM 1000 $x_1$ 5 14 0.2355 8.79E-08 16 69 0.0200 4.84E-06 2 9 0.0085 0.00E+00 $x_2$ 4 11 0.0156 1.06E-08 10 40 0.0126 5.44E-06 4 15 0.0079 0.00E+00 $x_3$ 15 35 0.0887 6.01E-06 13 54 0.1723 7.55E-06 5 20 0.0119 0.00E+00 $x_4$ 6 16 0.0204 5.47E-06 25 133 0.1284 3.35E-06 2 9 0.0079 0.00E+00 $x_5$ 8 20 0.0294 2.47E-06 27 138 0.0757 6.03E-06 2 11 0.0076 0.00E+00 $x_6$ 8 19 0.0362 1.95E-06 26 134 0.0350 4.52E-06 4 15 0.0102 0.00E+00 $x_7$ 8 20 0.0292 2.48E-06 25 133 0.1248 3.35E-06 2 11 0.0071 0.00E+00 $x_8$ 8 20 0.0299 2.49E-06 22 112 0.0317 4.53E-06 2 11 0.0080 0.00E+00 $x_1$ 5 14 0.0720 1.71E-07 17 73 0.0820 2.67E-06 2 9 0.0143 0.00E+00 $x_2$ 4 11 0.0457 2.31E-08 8 31 0.0563 8.85E-06 4 15 0.0182 0.00E+00 $x_3$ 15 35 0.0707 6.05E-06 13 54 1.5146 7.55E-06 5 20 0.0308 0.00E+00 $x_4$ 6 16 0.0492 5.61E-06 24 126 0.1178 2.59E-06 2 9 0.0154 0.00E+00 $x_5$ 8 20 0.0611 5.54E-06 39 226 0.4130 3.08E-06 2 11 0.0190 0.00E+00 $x_6$ 8 19 0.0831 1.95E-06 22 99 0.1434 5.73E-06 4 15 0.0251 0.00E+00 $x_7$ 8 20 0.0651 5.54E-06 24 126 0.1656 2.59E-06 2 11 0.0175 0.00E+00 $x_8$ 8 20 0.0476 5.55E-06 35 216 0.3063 8.62E-06 2 11 0.0242 0.00E+00 $x_1$ 5 14 0.0956 2.37E-07 15 60 0.1038 4.70E-06 2 9 0.0256 0.00E+00 $x_2$ 4 11 0.0494 3.25E-08 9 34 0.0697 5.29E-06 4 15 0.0457 0.00E+00 $x_3$ 15 35 0.2145 6.05E-06 13 54 3.1315 7.55E-06 5 20 0.0614 0.00E+00 $x_4$ 6 16 0.0723 5.75E-06 52 336 0.7500 6.45E-06 2 9 0.0262 0.00E+00 $x_5$ 8 20 0.1135 7.84E-06 39 257 0.8268 7.40E-06 2 11 0.0234 0.00E+00 $x_6$ 8 19 0.1167 1.95E-06 21 97 0.1880 7.11E-06 4 15 0.0360 0.00E+00 $x_7$ 8 20 0.1115 7.84E-06 58 456 0.8633 8.74E-06 2 11 0.0253 0.00E+00 $x_8$ 8 20 0.1302 7.84E-06 40 250 0.3797 4.87E-06 2 11 0.0324 0.00E+00 $x_1$ 7 23 0.3971 9.16E-11 14 56 0.2812 4.32E-06 3 16 0.1857 0.00E+00 $x_2$ 4 11 0.1609 7.25E-08 10 38 0.3246 3.24E-06 4 15 0.1384 0.00E+00 $x_3$ 15 35 0.6173 6.05E-06 13 54 25.8148 7.55E-06 5 20 0.1416 0.00E+00 $x_4$ 7 23 0.4199 7.98E-10 41 287 2.6315 4.61E-06 2 13 0.1230 0.00E+00 $x_5$ 9 23 0.3295 2.21E-06 43 261 7.0197 8.15E-06 2 10 0.0809 0.00E+00 $x_6$ 8 19 0.4273 1.95E-06 23 121 0.3488 2.56E-06 4 15 0.1403 0.00E+00 $x_7$ 9 23 0.4588 2.21E-06 42 295 2.2382 7.02E-06 3 13 0.1398 0.00E+00 $x_8$ 9 23 0.3960 2.21E-06 41 252 2.0126 5.39E-07 3 13 0.1062 0.00E+00 $x_1$ 7 27 0.9775 3.55E-07 14 56 0.5772 4.93E-06 3 18 0.5132 0.00E+00 $x_2$ 4 11 0.4880 1.03E-07 10 38 0.5872 4.73E-06 4 15 0.2445 0.00E+00 $x_3$ 15 35 1.0573 6.05E-06 13 54 87.3280 7.55E-06 5 20 0.3240 0.00E+00 $x_4$ 7 27 0.5609 3.63E-06 41 265 3.9792 2.29E-06 3 18 0.4003 0.00E+00 $x_5$ 9 24 1.1510 4.99E-06 52 420 13.2180 1.53E-06 2 10 0.2150 0.00E+00 $x_6$ 11 25 0.7559 9.20E-08 21 99 1.2739 4.64E-06 4 15 0.2824 0.00E+00 $x_7$ 9 24 1.0529 4.99E-06 41 266 3.6878 6.90E-06 3 13 0.1655 0.00E+00 $x_8$ 9 24 0.8091 4.99E-06 42 316 15.6351 3.58E-06 3 13 0.2459 0.00E+00

Table 3.  Numerical Results for DPPM, MDYP and WYLP for Problem 2 with given initial points and dimension, $f$ represents failure

 DPPM MDYP WYLP DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM 1000 $x_1$ 9 26 0.0452 4.37E-06 13 40 0.0794 3.39E-06 19 93 0.3206 9.75E-07 $x_2$ 7 19 0.0244 7.69E-06 7 22 0.0300 5.13E-06 15 73 0.0934 4.17E-07 $x_3$ 8 22 0.0703 3.51E-06 11 36 0.1053 3.48E-06 15 73 0.1950 4.96E-07 $x_4$ 9 26 0.0401 6.71E-06 14 46 0.0217 7.56E-06 19 93 0.0969 9.62E-07 $x_5$ 10 29 0.0461 3.13E-06 14 46 0.0238 7.56E-06 19 93 0.1020 4.80E-07 $x_6$ 9 25 0.0377 6.09E-06 10 32 0.0135 8.15E-06 16 78 0.0441 6.75E-07 $x_7$ 10 29 0.0764 3.13E-06 14 46 0.0165 7.56E-06 19 93 0.0604 4.80E-07 $x_8$ 10 29 0.0419 3.13E-06 14 46 0.0264 7.53E-06 19 93 0.0568 4.81E-07 $x_1$ 9 26 0.1909 9.84E-06 13 40 0.0641 7.50E-06 20 98 0.4133 8.51E-07 $x_2$ 8 22 0.0677 3.39E-06 8 25 0.0360 4.94E-06 15 73 0.1504 9.03E-07 $x_3$ 8 22 0.1055 3.51E-06 11 36 1.2062 8.57E-06 15 73 0.1302 4.89E-07 $x_4$ 10 29 0.1083 2.25E-06 35 157 0.1346 7.40E-06 20 98 0.1267 8.49E-07 $x_5$ 10 29 0.1217 7.01E-06 31 138 0.4007 4.24E-06 20 98 0.1494 4.20E-07 $x_6$ 9 25 0.0721 6.02E-06 10 32 0.0487 8.13E-06 16 78 0.1823 6.64E-07 $x_7$ 10 29 0.1134 7.01E-06 31 138 0.2469 4.24E-06 20 98 0.1852 4.20E-07 $x_8$ 10 29 0.0780 7.01E-06 22 88 0.3009 7.25E-06 20 98 0.1700 4.21E-07 $x_1$ 10 29 0.1738 2.79E-06 14 43 0.1073 2.89E-06 21 103 0.5148 4.80E-07 $x_2$ 8 22 0.0853 4.78E-06 8 25 0.0799 6.92E-06 16 78 0.2554 5.09E-07 $x_3$ 8 22 0.0803 3.51E-06 11 36 3.0274 9.89E-06 15 73 0.1792 4.88E-07 $x_4$ 10 29 0.1861 2.99E-06 16 54 0.4752 4.83E-06 21 103 0.3229 4.79E-07 $x_5$ 10 29 0.2444 9.91E-06 16 54 0.1676 4.83E-06 20 98 0.3376 5.93E-07 $x_6$ 9 25 0.2425 6.02E-06 10 32 0.0720 8.13E-06 16 78 0.2158 6.63E-07 $x_7$ 10 29 0.2000 9.91E-06 16 54 0.1314 4.83E-06 20 98 0.2923 5.93E-07 $x_8$ 10 29 0.2440 9.92E-06 16 54 0.0871 4.69E-06 20 98 0.3689 5.93E-07 $x_1$ 12 39 1.0114 2.30E-06 14 43 0.4292 6.46E-06 23 116 2.6259 6.55E-07 $x_2$ 9 25 0.6191 2.13E-06 9 28 0.3785 1.05E-06 17 83 0.9079 4.53E-07 $x_3$ 8 22 0.3316 3.51E-06 12 39 24.3728 3.64E-06 15 73 0.4278 4.88E-07 $x_4$ 12 39 0.6189 2.30E-06 f f f f 23 116 1.4482 6.55E-07 $x_5$ 11 32 0.8394 4.43E-06 f f f f 21 103 1.7160 5.29E-07 $x_6$ 9 25 0.5946 6.01E-06 10 32 1.1865 8.12E-06 16 78 0.7065 6.62E-07 $x_7$ 11 32 0.7496 4.43E-06 f f f f 21 103 1.1586 5.29E-07 $x_8$ 11 32 0.8790 4.44E-06 f f f f 21 103 0.8051 5.29E-07 $x_1$ 12 45 2.1717 2.90E-06 14 43 1.1046 9.13E-06 25 130 3.6530 6.38E-07 $x_2$ 9 25 1.3327 3.02E-06 9 28 0.7541 1.49E-06 17 81 1.5444 1.60E-11 $x_3$ 8 22 0.9087 3.51E-06 12 41 94.8845 3.66E-06 15 73 1.2575 4.88E-07 $x_4$ 12 45 1.7092 3.05E-06 f f f f 25 130 1.9217 6.38E-07 $x_5$ 13 40 1.2000 2.81E-06 f f f f 22 110 2.0242 6.94E-07 $x_6$ 9 25 1.2705 6.01E-06 10 32 0.8082 8.12E-06 16 78 1.4797 6.62E-07 $x_7$ 13 40 1.9988 2.81E-06 f f f f 22 110 1.7161 6.94E-07 $x_8$ 13 40 1.7253 2.81E-06 f f f f 22 110 2.3099 6.94E-07

Table 4.  Numerical Results for DPPM, MDYP and WYLP for Problem 3 with given initial points and dimension, $f$ represents failure

 DPPM MDYP WYLP DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM 1000 $x_1$ 4 8 0.8195 6.47E-09 12 39 0.0871 9.82E-06 2 8 0.0309 0.00E+00 $x_2$ 3 6 0.0912 8.25E-08 10 32 0.0564 9.57E-06 2 8 0.0068 0.00E+00 $x_3$ 13 27 0.0953 8.81E-06 9 29 0.2011 2.35E-06 13 51 0.0271 3.85E-07 $x_4$ 4 8 0.0192 1.42E-06 37 210 0.1077 2.98E-06 17 69 0.0250 7.21E-07 $x_5$ 6 12 0.0705 5.56E-06 37 210 0.0689 2.98E-06 17 73 0.0373 8.83E-07 $x_6$ 5 10 0.0919 6.68E-11 16 56 0.0211 7.32E-06 15 59 0.0292 8.78E-07 $x_7$ 6 12 0.0203 5.56E-06 37 210 0.0639 2.98E-06 17 73 0.0332 8.83E-07 $x_8$ 8 16 0.1217 4.35E-08 19 85 0.0470 6.92E-06 17 73 0.0263 8.83E-07 $x_1$ 4 8 0.0746 1.45E-08 13 42 0.0529 3.54E-06 2 8 0.0133 0.00E+00 $x_2$ 3 6 0.0249 1.84E-07 11 35 0.0574 3.71E-06 2 8 0.0097 0.00E+00 $x_3$ 13 27 0.1299 8.81E-06 9 29 1.1781 2.35E-06 13 51 0.0781 3.85E-07 $x_4$ 4 8 0.0397 1.28E-06 36 209 0.4912 3.49E-06 18 72 0.0891 3.15E-07 $x_5$ 13 26 0.0749 7.27E-13 36 209 0.3499 3.49E-06 18 76 0.1789 6.06E-07 $x_6$ 5 10 0.0320 6.70E-11 21 81 0.1014 7.63E-06 16 62 0.0643 9.17E-07 $x_7$ 13 26 0.0643 7.27E-13 36 209 0.3509 3.49E-06 18 76 0.1192 6.08E-07 $x_8$ 9 18 0.0524 8.51E-09 20 84 0.1209 8.44E-06 18 76 0.0923 6.07E-07 $x_1$ 4 8 0.3372 2.04E-08 13 42 0.0841 5.00E-06 2 8 0.0186 0.00E+00 $x_2$ 3 6 0.0404 2.61E-07 11 35 0.0801 5.24E-06 2 8 0.0165 0.00E+00 $x_3$ 13 27 0.1017 8.81E-06 9 29 3.2921 2.35E-06 13 51 0.0892 3.85E-07 $x_4$ 4 8 0.0452 1.30E-06 31 177 0.4489 1.82E-06 15 59 0.0737 8.90E-07 $x_5$ 11 22 0.1231 1.43E-10 31 177 0.3576 1.82E-06 18 76 0.0882 8.57E-07 $x_6$ 5 10 0.0408 6.70E-11 18 72 0.1468 8.19E-06 f f f f $x_7$ 11 22 0.0840 1.43E-10 31 177 0.3460 1.82E-06 18 76 0.1498 8.57E-07 $x_8$ 12 24 0.0726 1.21E-11 29 153 0.3801 6.17E-06 18 76 0.1678 8.55E-07 $x_1$ 5 15 0.1876 1.58E-06 14 45 0.3743 4.52E-06 3 14 0.2255 0.00E+00 $x_2$ 3 6 0.1362 5.83E-07 12 38 0.3290 4.63E-06 2 8 0.1014 0.00E+00 $x_3$ 13 27 0.4447 8.81E-06 9 29 22.4652 2.35E-06 13 51 0.3770 3.85E-07 $x_4$ 6 17 0.3568 2.93E-10 40 278 8.4671 7.10E-06 18 73 0.6564 6.79E-07 $x_5$ 17 36 0.6444 7.27E-06 40 278 9.2475 7.10E-06 f f f f $x_6$ 5 10 0.1576 6.71E-11 21 80 4.8421 9.53E-06 f f f f $x_7$ 17 36 0.4210 7.27E-06 40 278 8.9672 7.10E-06 f f f f $x_8$ 17 36 0.4061 7.27E-06 36 222 0.7206 4.06E-06 f f f f $x_1$ 6 21 0.9043 8.13E-09 14 45 0.3008 6.39E-06 4 21 0.2761 0.00E+00 $x_2$ 3 6 0.2392 8.25E-07 12 38 0.6761 6.55E-06 2 8 0.1008 0.00E+00 $x_3$ 13 27 0.5491 8.81E-06 9 29 72.1280 2.35E-06 13 51 0.5589 3.85E-07 $x_4$ 6 21 0.4396 1.81E-08 34 224 3.5081 4.61E-06 19 80 1.1189 6.33E-07 $x_5$ 18 40 1.4053 7.38E-06 34 224 2.4146 4.61E-06 f f f f $x_6$ 5 10 0.3521 6.71E-11 19 71 7.7563 4.00E-06 f f f f $x_7$ 18 40 1.0043 7.38E-06 34 224 3.4369 4.61E-06 f f f f $x_8$ 18 40 1.2596 7.38E-06 37 225 3.3577 4.58E-06 f f f f

Table 5.  Numerical Results for DPPM, MDYP and WYLP for Problem 4 with given initial points and dimension, $f$ represents failure

 DPPM MDYP WYLP DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM 1000 $x_1$ 10 29 0.1975 5.56E-06 13 41 0.0277 2.04E-06 17 83 0.0574 5.78E-07 $x_2$ 10 29 0.0528 8.48E-06 13 41 0.0191 3.11E-06 19 89 0.0732 7.34E-07 $x_3$ 10 28 0.0530 1.66E-14 13 41 0.0182 3.22E-06 f f f f $x_4$ 10 28 0.0521 1.31E-14 13 41 0.0229 2.65E-06 f f f f $x_5$ 10 28 0.0362 1.23E-14 13 41 0.0206 2.65E-06 20 92 0.0418 6.37E-07 $x_6$ 10 28 0.0370 1.12E-14 13 41 0.0291 3.22E-06 17 83 0.0544 7.43E-07 $x_7$ 10 28 0.0345 1.23E-14 13 41 0.0273 2.65E-06 20 92 0.0623 6.37E-07 $x_8$ 10 28 0.0388 1.16E-14 13 41 0.0181 2.65E-06 20 92 0.0504 6.37E-07 $x_1$ 11 32 0.1438 2.49E-06 13 41 0.2237 4.56E-06 f f f f $x_2$ 10 30 0.1277 0.00E+00 13 41 0.2464 6.96E-06 657 2006 2.2801 4.71E-07 $x_3$ 11 34 0.1592 9.60E-06 13 41 0.0676 7.22E-06 21 104 0.2050 3.40E-08 $x_4$ 11 32 0.1546 2.49E-06 13 41 0.0814 5.94E-06 17 83 0.1518 8.10E-07 $x_5$ 11 34 0.0902 7.90E-06 13 41 0.1253 5.94E-06 21 105 0.1327 6.96E-07 $x_6$ 11 34 0.2370 9.60E-06 13 41 0.0804 7.22E-06 19 94 0.1391 1.91E-07 $x_7$ 11 34 0.1372 7.90E-06 13 41 0.0817 5.94E-06 21 105 0.1985 6.96E-07 $x_8$ 11 34 0.1759 7.90E-06 13 41 0.0618 5.94E-06 21 105 0.1921 6.96E-07 $x_1$ 10 30 0.3047 8.88E-16 13 41 0.1117 6.46E-06 248 777 2.4396 7.71E-07 $x_2$ 12 41 0.4791 8.80E-06 13 41 0.2053 9.84E-06 321 999 3.1750 7.52E-07 $x_3$ 11 37 0.2859 1.33E-15 14 44 0.2424 4.02E-06 f f f f $x_4$ 11 34 0.2346 8.59E-06 13 41 0.1677 8.41E-06 27 116 0.1964 4.43E-07 $x_5$ 12 39 0.3669 4.87E-06 13 41 0.1120 8.41E-06 22 111 0.4084 6.30E-07 $x_6$ 12 41 0.2340 9.13E-06 14 44 0.1076 4.02E-06 240 756 2.1246 6.20E-07 $x_7$ 12 39 0.3153 4.87E-06 13 41 0.1737 8.41E-06 22 111 0.5055 6.30E-07 $x_8$ 12 39 0.3989 4.87E-06 13 41 0.1488 8.40E-06 22 111 0.3394 6.30E-07 $x_1$ 9 38 0.8360 0.00E+00 14 44 0.6011 5.69E-06 22 107 1.0331 4.71E-07 $x_2$ 13 66 1.4044 9.93E-14 14 44 0.7478 8.67E-06 23 121 1.4131 8.76E-07 $x_3$ 13 69 1.1444 9.93E-16 14 44 0.7075 9.00E-06 29 140 1.7453 9.92E-07 $x_4$ 10 41 1.0490 2.03E-14 14 44 0.5255 7.40E-06 f f f f $x_5$ 11 53 1.4307 4.86E-14 14 44 0.6108 7.40E-06 277 878 5.8666 8.74E-07 $x_6$ 14 72 1.6761 7.15E-15 14 44 0.6519 9.00E-06 f f f f $x_7$ 11 53 1.0887 4.86E-14 14 44 0.5013 7.40E-06 277 878 5.8247 8.74E-07 $x_8$ 11 53 0.9404 4.86E-14 14 44 0.6483 7.40E-06 23 116 1.6068 8.75E-07 $x_1$ 13 61 2.9900 1.40E-13 14 44 1.2238 8.04E-06 21 114 2.6706 4.32E-07 $x_2$ 15 92 4.1893 1.40E-13 15 47 1.0116 2.14E-06 24 140 3.1769 7.35E-07 $x_3$ 15 98 3.8911 1.40E-13 15 47 1.2457 2.23E-06 29 158 2.5033 8.14E-07 $x_4$ 11 55 2.2741 1.72E-14 15 47 1.3337 1.83E-06 21 114 1.5601 6.03E-07 $x_5$ 13 77 3.0852 9.15E-14 15 47 1.4839 1.83E-06 75 284 7.6840 8.99E-07 $x_6$ 15 98 3.7006 1.33E-13 15 47 1.2636 2.23E-06 567 1772 31.9028 7.18E-07 $x_7$ 13 77 3.3259 9.15E-14 15 47 1.3969 1.83E-06 75 284 5.1793 8.99E-07 $x_8$ 13 77 2.3221 9.15E-14 15 47 1.4001 1.83E-06 75 284 5.6734 8.99E-07

Table 6.  Numerical Results for DPPM, MDYP and WYLP for Problem 5 with given initial points and dimension, $f$ represents failure

 DPPM MDYP WYLP DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM 1000 $x_1$ 5 12 0.2244 3.26E-06 11 38 0.2095 3.21E-06 2 9 0.0260 0.00E+00 $x_2$ 4 8 0.1030 7.57E-08 9 29 0.0294 8.71E-06 2 8 0.0143 0.00E+00 $x_3$ 5 11 0.0745 1.28E-07 12 38 0.1401 1.70E-06 4 14 0.0322 2.22E-16 $x_4$ 7 16 0.0238 1.92E-06 21 79 0.0319 6.66E-06 12 45 0.0305 7.78E-08 $x_5$ 9 19 0.0954 6.02E-06 21 79 0.0333 6.66E-06 14 54 0.0385 3.41E-07 $x_6$ 9 19 0.0316 4.09E-06 11 36 0.0131 3.32E-06 14 54 0.0249 2.67E-07 $x_7$ 9 19 0.0215 6.02E-06 21 79 0.0265 6.66E-06 14 54 0.0204 3.42E-07 $x_8$ 9 19 0.0286 5.97E-06 17 61 0.0164 6.86E-06 14 54 0.0198 3.92E-07 $x_1$ 5 12 0.0353 7.29E-06 11 38 0.0340 7.19E-06 2 9 0.0136 0.00E+00 $x_2$ 4 8 0.0247 1.69E-07 10 32 0.0334 8.33E-06 2 8 0.0197 0.00E+00 $x_3$ 5 11 0.0330 1.28E-07 12 38 1.4578 1.70E-06 4 14 0.0172 2.22E-16 $x_4$ 7 16 0.0494 1.79E-06 19 69 0.0617 2.22E-06 12 47 0.0369 9.52E-07 $x_5$ 9 20 0.0565 2.04E-06 19 69 0.0724 2.22E-06 15 59 0.0548 3.37E-07 $x_6$ 9 19 0.0677 4.92E-06 11 36 0.0424 3.32E-06 14 54 0.0385 6.25E-07 $x_7$ 9 20 0.0583 2.04E-06 19 69 0.0908 2.22E-06 15 59 0.0642 3.32E-07 $x_8$ 9 20 0.0440 2.04E-06 27 122 0.1471 3.72E-06 15 59 0.0605 3.36E-07 $x_1$ 6 14 0.0693 3.67E-10 12 41 0.0766 4.72E-06 2 9 0.0312 0.00E+00 $x_2$ 4 8 0.0665 2.39E-07 11 35 0.0800 1.49E-06 2 8 0.0131 0.00E+00 $x_3$ 5 11 0.0537 1.28E-07 12 38 2.5950 1.70E-06 4 14 0.0239 2.22E-16 $x_4$ 8 18 0.1273 3.07E-07 17 62 0.1395 9.13E-06 12 47 0.0824 7.20E-07 $x_5$ 9 20 0.0858 2.89E-06 17 62 0.1122 9.13E-06 15 59 0.0614 4.78E-07 $x_6$ 9 19 0.1316 4.92E-06 11 36 0.0749 3.32E-06 14 54 0.0617 7.36E-07 $x_7$ 9 20 0.0844 2.89E-06 17 62 0.1038 9.13E-06 15 59 0.0613 4.75E-07 $x_8$ 9 20 0.1005 2.89E-06 27 103 0.4661 2.52E-06 15 59 0.0524 4.75E-07 $x_1$ 7 20 0.4618 3.27E-08 13 44 0.3079 7.07E-07 2 10 0.0474 0.00E+00 $x_2$ 4 8 0.1123 5.35E-07 11 35 0.3012 3.33E-06 2 8 0.0923 0.00E+00 $x_3$ 5 11 0.1031 1.28E-07 12 38 23.8270 1.70E-06 4 14 0.1142 2.22E-16 $x_4$ 7 20 0.4022 3.43E-08 18 65 0.5567 8.62E-06 10 42 0.2322 7.89E-08 $x_5$ 9 21 0.3592 4.85E-06 18 65 0.3972 8.62E-06 16 65 0.4585 8.55E-07 $x_6$ 9 19 0.3959 4.92E-06 11 36 0.2964 3.32E-06 f f f f $x_7$ 9 21 0.3742 4.85E-06 18 65 0.4319 8.62E-06 16 65 0.1987 8.49E-07 $x_8$ 9 21 0.3703 4.85E-06 18 65 0.5524 8.97E-06 16 65 0.5407 8.39E-07 $x_1$ 7 23 0.7369 4.77E-06 13 44 0.7618 1.00E-06 3 16 0.1932 0.00E+00 $x_2$ 4 8 0.2658 7.57E-07 11 35 0.4434 4.71E-06 2 8 0.1158 0.00E+00 $x_3$ 5 11 0.3353 1.28E-07 12 38 107.4606 1.70E-06 4 14 0.2615 2.22E-16 $x_4$ 7 23 0.7591 4.81E-06 19 68 0.4613 2.35E-06 14 59 0.6242 5.49E-07 $x_5$ 9 22 0.7743 8.65E-06 19 68 1.1533 2.35E-06 f f f f $x_6$ 9 19 0.6254 4.92E-06 11 36 0.5491 3.32E-06 f f f f $x_7$ 9 22 0.7055 8.65E-06 19 68 1.0239 2.35E-06 f f f f $x_8$ 9 22 0.5172 8.66E-06 19 68 0.9601 2.41E-06 f f f f

Figures(3)

Tables(6)