1 Introduction
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The search direction has a sufficient descent feature and a trust region trait.
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Under mild assumptions, the proposed algorithm possesses the global convergence.
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The new algorithm combines the steepest descent method with the conjugate gradient algorithm.
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Numerical results prove that it is perfect compared to other similar algorithms.
2 New modified conjugate gradient algorithm
3 Important characteristics
4 Numerical results
4.1 Problems and test experiments
No. | Problem |
---|---|
1 | Extended Freudenstein and Roth Function |
2 | Extended Trigonometric Function |
3 | Extended Rosenbrock Function |
4 | Extended White and Holst Function |
5 | Extended Beale Function |
6 | Extended Penalty Function |
7 | Perturbed Quadratic Function |
8 | Raydan 1 Function |
9 | Raydan 2 Function |
10 | Diagonal 1 Function |
11 | Diagonal 2 Function |
12 | Diagonal 3 Function |
13 | Hager Function |
14 | Generalized Tridiagonal 1 Function |
15 | Extended Tridiagonal 1 Function |
16 | Extended Three Exponential Terms Function |
17 | Generalized Tridiagonal 2 Function |
18 | Diagonal 4 Function |
19 | Diagonal 5 Function |
20 | Extended Himmelblau Function |
21 | Generalized PSC1 Function |
22 | Extended PSC1 Function |
23 | Extended Powell Function |
24 | Extended Block Diagonal BD1 Function |
25 | Extended Maratos Function |
26 | Extended Cliff Function |
27 | Quadratic Diagonal Perturbed Function |
28 | Extended Wood Function |
29 | Extended Hiebert Function |
30 | Quadratic Function QF1 Function |
31 | Extended Quadratic Penalty QP1 Function |
32 | Extended Quadratic Penalty QP2 Function |
33 | A Quadratic Function QF2 Function |
34 | Extended EP1 Function |
35 | Extended Tridiagonal-2 Function |
36 | BDQRTIC Function (CUTE) |
37 | TRIDIA Function (CUTE) |
38 | ARWHEAD Function (CUTE) |
38 | ARWHEAD Function (CUTE) |
40 | NONDQUAR Function (CUTE) |
41 | DQDRTIC Function (CUTE) |
42 | EG2 Function (CUTE) |
43 | DIXMAANA Function (CUTE) |
44 | DIXMAANB Function (CUTE) |
45 | DIXMAANC Function (CUTE) |
46 | DIXMAANE Function (CUTE) |
47 | Partial Perturbed Quadratic Function |
48 | Broyden Tridiagonal Function |
49 | Almost Perturbed Quadratic Function |
50 | Tridiagonal Perturbed Quadratic Function |
51 | EDENSCH Function (CUTE) |
52 | VARDIM Function (CUTE) |
53 | STAIRCASE S1 Function |
54 | LIARWHD Function (CUTE) |
55 | DIAGONAL 6 Function |
56 | DIXON3DQ Function (CUTE) |
57 | DIXMAANF Function (CUTE) |
58 | DIXMAANG Function (CUTE) |
59 | DIXMAANH Function (CUTE) |
60 | DIXMAANI Function (CUTE) |
61 | DIXMAANJ Function (CUTE) |
62 | DIXMAANK Function (CUTE) |
63 | DIXMAANL Function (CUTE) |
64 | DIXMAAND Function (CUTE) |
65 | ENGVAL1 Function (CUTE) |
66 | FLETCHCR Function (CUTE) |
67 | COSINE Function (CUTE) |
68 | Extended DENSCHNB Function (CUTE) |
69 | DENSCHNF Function (CUTE) |
70 | SINQUAD Function (CUTE) |
71 | BIGGSB1 Function (CUTE) |
72 | Partial Perturbed Quadratic PPQ2 Function |
73 | Scaled Quadratic SQ1 Function |
NO | Dim | Algorithm 2.1 | Algorithm 2 | Algorithm 3 | ||||||
---|---|---|---|---|---|---|---|---|---|---|
NI | NFG | CPU | NI | NFG | CPU | NI | NFG | CPU | ||
1 | 9000 | 4 | 20 | 0.124801 | 14 | 48 | 0.405603 | 5 | 26 | 0.249602 |
2 | 9000 | 71 | 327 | 1.965613 | 27 | 89 | 0.670804 | 32 | 136 | 0.858005 |
3 | 9000 | 7 | 20 | 0.0312 | 37 | 160 | 0.249602 | 27 | 147 | 0.202801 |
4 | 9000 | 12 | 49 | 0.280802 | 34 | 161 | 0.717605 | 42 | 219 | 0.951606 |
5 | 9000 | 13 | 56 | 0.202801 | 20 | 63 | 0.249602 | 5 | 24 | 0.0624 |
6 | 9000 | 65 | 252 | 0.421203 | 43 | 143 | 0.280802 | 3 | 9 | 0.0312 |
7 | 9000 | 11 | 37 | 0.0624 | 478 | 979 | 2.215214 | 465 | 1479 | 2.558416 |
8 | 9000 | 5 | 20 | 0.0624 | 22 | 55 | 0.156001 | 14 | 54 | 0.156001 |
9 | 9000 | 6 | 16 | 0.0312 | 5 | 21 | 0.0624 | 3 | 8 | 0.0312 |
10 | 9000 | 2 | 13 | 0.0156 | 2 | 13 | 0.000001 | 2 | 13 | 0.000001 |
11 | 9000 | 3 | 17 | 0.0312 | 7 | 34 | 0.0624 | 17 | 87 | 0.218401 |
12 | 9000 | 3 | 10 | 0.0312 | 19 | 40 | 0.202801 | 14 | 50 | 0.202801 |
13 | 9000 | 3 | 24 | 0.0624 | 3 | 24 | 0.0312 | 3 | 24 | 0.0156 |
14 | 9000 | 4 | 12 | 4.305628 | 5 | 14 | 5.382034 | 5 | 14 | 5.226033 |
15 | 9000 | 19 | 77 | 9.984064 | 22 | 66 | 9.516061 | 21 | 71 | 10.296066 |
16 | 9000 | 3 | 11 | 0.0624 | 6 | 27 | 0.078 | 6 | 18 | 0.0624 |
17 | 9000 | 11 | 45 | 0.374402 | 27 | 69 | 0.780005 | 27 | 87 | 0.811205 |
18 | 9000 | 5 | 23 | 0.0312 | 3 | 10 | 0.000001 | 3 | 10 | 0.0312 |
19 | 9000 | 3 | 9 | 0.0624 | 3 | 9 | 0.0312 | 3 | 19 | 0.0312 |
20 | 9000 | 19 | 76 | 0.124801 | 15 | 36 | 0.0624 | 3 | 9 | 0.0312 |
21 | 9000 | 12 | 47 | 0.156001 | 13 | 61 | 0.187201 | 15 | 59 | 0.218401 |
22 | 9000 | 7 | 46 | 0.795605 | 8 | 70 | 0.577204 | 6 | 46 | 0.686404 |
23 | 9000 | 9 | 45 | 0.218401 | 101 | 357 | 2.090413 | 46 | 150 | 0.873606 |
24 | 9000 | 5 | 47 | 0.093601 | 14 | 88 | 0.156001 | 14 | 97 | 0.249602 |
25 | 9000 | 9 | 28 | 0.0312 | 40 | 214 | 0.249602 | 8 | 46 | 0.0624 |
26 | 9000 | 24 | 102 | 0.327602 | 24 | 100 | 0.249602 | 3 | 24 | 0.0312 |
27 | 9000 | 6 | 20 | 0.0312 | 34 | 109 | 0.187201 | 92 | 321 | 0.530403 |
28 | 9000 | 13 | 50 | 0.124801 | 20 | 83 | 0.109201 | 23 | 84 | 0.140401 |
29 | 9000 | 6 | 36 | 0.0468 | 4 | 21 | 0.0312 | 4 | 21 | 0.0312 |
30 | 9000 | 11 | 37 | 0.0624 | 454 | 931 | 1.450809 | 424 | 1346 | 1.747211 |
31 | 9000 | 18 | 63 | 0.124801 | 15 | 51 | 0.093601 | 3 | 10 | 0.0312 |
32 | 9000 | 18 | 70 | 0.218401 | 23 | 61 | 0.218401 | 3 | 18 | 0.0624 |
33 | 9000 | 2 | 5 | 0.000001 | 2 | 5 | 0.0312 | 2 | 5 | 0.000001 |
34 | 9000 | 8 | 16 | 0.0312 | 6 | 12 | 0.0312 | 3 | 6 | 0.0312 |
35 | 9000 | 4 | 13 | 0.0312 | 4 | 10 | 0.0312 | 3 | 8 | 0.000001 |
36 | 9000 | 7 | 23 | 4.602029 | 8 | 28 | 5.569236 | 10 | 47 | 8.673656 |
37 | 9000 | 7 | 23 | 0.0624 | 1412 | 2829 | 6.942044 | 2000 | 6021 | 11.356873 |
38 | 9000 | 4 | 18 | 0.0312 | 8 | 35 | 0.187201 | 4 | 11 | 0.0312 |
39 | 9000 | 5 | 19 | 0.0312 | 28 | 56 | 0.124801 | 3 | 8 | 0.0312 |
40 | 9000 | 13 | 43 | 0.561604 | 835 | 2936 | 36.223432 | 9 | 41 | 0.421203 |
41 | 9000 | 10 | 32 | 0.0624 | 17 | 41 | 0.093601 | 22 | 81 | 0.124801 |
42 | 9000 | 4 | 33 | 0.0624 | 13 | 35 | 0.124801 | 9 | 47 | 0.109201 |
43 | 9000 | 16 | 62 | 1.029607 | 16 | 38 | 0.951606 | 13 | 48 | 0.780005 |
44 | 9000 | 3 | 17 | 0.156001 | 9 | 50 | 0.624004 | 3 | 17 | 0.187201 |
45 | 9000 | 21 | 118 | 1.49761 | 12 | 81 | 0.858006 | 3 | 24 | 0.202801 |
46 | 9000 | 20 | 81 | 1.435209 | 209 | 443 | 11.247672 | 110 | 362 | 6.630042 |
47 | 9000 | 11 | 37 | 27.066173 | 30 | 97 | 68.64044 | 37 | 112 | 87.220159 |
48 | 9000 | 13 | 54 | 9.718862 | 31 | 92 | 18.610919 | 23 | 50 | 11.980877 |
49 | 9000 | 11 | 37 | 0.0624 | 478 | 979 | 1.51321 | 504 | 1592 | 1.887612 |
50 | 9000 | 11 | 37 | 7.971651 | 472 | 967 | 263.68849 | 444 | 1273 | 299.381519 |
51 | 9000 | 6 | 31 | 0.156001 | 7 | 25 | 0.218401 | 3 | 17 | 0.124801 |
52 | 9000 | 62 | 186 | 0.998406 | 63 | 195 | 0.842405 | 4 | 21 | 0.0624 |
53 | 9000 | 10 | 32 | 0.0312 | 2000 | 4059 | 7.72205 | 1865 | 5618 | 7.971651 |
54 | 9000 | 4 | 11 | 0.0312 | 21 | 79 | 0.156001 | 17 | 79 | 0.124801 |
55 | 9000 | 10 | 24 | 3.010819 | 7 | 25 | 3.213621 | 3 | 10 | 1.076407 |
56 | 9000 | 7 | 21 | 0.0156 | 2000 | 4003 | 6.489642 | 1390 | 4107 | 5.335234 |
57 | 9000 | 5 | 39 | 0.358802 | 67 | 220 | 4.024826 | 3 | 24 | 0.202801 |
58 | 9000 | 5 | 24 | 0.343202 | 114 | 282 | 6.411641 | 82 | 315 | 5.257234 |
59 | 9000 | 5 | 39 | 0.343202 | 68 | 310 | 4.72683 | 3 | 23 | 0.171601 |
60 | 9000 | 18 | 74 | 1.294808 | 206 | 437 | 11.107271 | 119 | 363 | 6.957645 |
61 | 9000 | 5 | 39 | 0.358802 | 85 | 247 | 4.929632 | 3 | 24 | 0.218401 |
62 | 9000 | 4 | 32 | 0.234001 | 4 | 32 | 0.249602 | 3 | 22 | 0.187201 |
63 | 9000 | 3 | 22 | 0.187201 | 3 | 22 | 0.187201 | 3 | 22 | 0.187201 |
64 | 9000 | 5 | 39 | 0.343202 | 23 | 147 | 1.747211 | 3 | 23 | 0.218401 |
65 | 9000 | 12 | 59 | 15.334898 | 14 | 51 | 14.944896 | 7 | 21 | 6.130839 |
66 | 9000 | 3 | 9 | 1.62241 | 2000 | 4022 | 1114.767546 | 529 | 2196 | 443.526443 |
67 | 9000 | 5 | 28 | 0.093601 | 15 | 58 | 0.280802 | 3 | 23 | 0.0312 |
68 | 9000 | 13 | 55 | 0.109201 | 11 | 27 | 0.0624 | 9 | 25 | 0.0624 |
69 | 9000 | 16 | 73 | 0.218401 | 24 | 55 | 0.187201 | 20 | 70 | 0.171601 |
70 | 9000 | 4 | 13 | 2.542816 | 41 | 203 | 36.332633 | 35 | 231 | 37.783442 |
71 | 9000 | 11 | 35 | 0.093601 | 2000 | 4014 | 6.708043 | 1491 | 4631 | 5.600436 |
72 | 9000 | 9 | 30 | 21.85574 | 1089 | 3897 | 2675.588751 | 287 | 1015 | 704.391315 |
73 | 9000 | 19 | 65 | 0.093601 | 607 | 1269 | 1.856412 | 669 | 2062 | 2.293215 |
No. | Problem |
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1 | Exponential function 1 |
2 | Exponential function 2 |
3 | Trigonometric function |
4 | Singular function |
5 | Logarithmic function |
6 | Broyden tridiagonal function |
7 | Trigexp function |
8 | Strictly convex function 1 |
9 | Strictly convex function 2 |
10 | Zero Jacobian function |
11 | Linear function-full rank |
12 | Penalty function |
13 | Variable dimensioned function |
14 | Extended Powel singular function |
15 | Tridiagonal system |
16 | Five-diagonal system |
17 | Extended Freudentein and Roth function |
18 | Extended Wood problem |
19 | Discrete boundary value problem |
4.2 Results and discussion
5 Nonlinear equations
6 The global convergence of Algorithm 5.1
7 The results of nonlinear equations
7.1 Problems and test experiments
NO | Dim | Algorithm 5.1 | Algorithm 6 | ||||
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NI | NFG | CPU | NI | NFG | CPU | ||
1 | 3000 | 161 | 162 | 3.931225 | 146 | 147 | 4.149627 |
1 | 6000 | 126 | 127 | 12.760882 | 115 | 116 | 11.122871 |
1 | 9000 | 111 | 112 | 22.464144 | 99 | 100 | 19.515725 |
2 | 3000 | 5 | 76 | 1.185608 | 5 | 76 | 1.060807 |
2 | 6000 | 6 | 91 | 4.758031 | 5 | 76 | 4.009226 |
2 | 9000 | 5 | 62 | 6.926444 | 5 | 62 | 6.754843 |
3 | 3000 | 33 | 228 | 3.276021 | 18 | 106 | 1.778411 |
3 | 6000 | 40 | 275 | 15.490899 | 18 | 106 | 6.084039 |
3 | 9000 | 40 | 285 | 33.243813 | 18 | 106 | 12.54248 |
4 | 3000 | 4 | 61 | 0.842405 | 4 | 61 | 0.936006 |
4 | 6000 | 4 | 47 | 2.698817 | 4 | 61 | 3.322821 |
4 | 9000 | 4 | 47 | 5.226033 | 4 | 61 | 6.817244 |
5 | 3000 | 23 | 237 | 3.244821 | 23 | 237 | 3.354022 |
5 | 6000 | 25 | 263 | 14.133691 | 25 | 263 | 13.930889 |
5 | 9000 | 26 | 278 | 30.186193 | 26 | 278 | 30.092593 |
6 | 3000 | 1999 | 29986 | 382.951255 | 1999 | 29986 | 365.369942 |
6 | 6000 | 88 | 1307 | 68.141237 | 1999 | 29986 | 1484.240314 |
6 | 9000 | 65 | 962 | 101.806253 | 1999 | 29986 | 3113.998361 |
7 | 3000 | 4 | 47 | 0.748805 | 3 | 46 | 0.624004 |
7 | 6000 | 4 | 47 | 2.589617 | 3 | 46 | 2.386815 |
7 | 9000 | 4 | 47 | 5.257234 | 3 | 46 | 5.054432 |
8 | 3000 | 25 | 156 | 2.854818 | 17 | 142 | 1.872012 |
8 | 6000 | 32 | 189 | 10.826469 | 18 | 162 | 8.377254 |
8 | 9000 | 28 | 192 | 21.512538 | 19 | 174 | 18.938521 |
9 | 3000 | 10 | 151 | 1.934412 | 5 | 76 | 1.014007 |
9 | 6000 | 4 | 61 | 3.510023 | 5 | 76 | 3.884425 |
9 | 9000 | 4 | 61 | 6.614442 | 6 | 91 | 9.609662 |
10 | 3000 | 1999 | 29986 | 386.804479 | 1999 | 29986 | 359.816306 |
10 | 6000 | 1999 | 29986 | 1523.068963 | 1999 | 29986 | 1469.59182 |
10 | 9000 | 1999 | 29986 | 3164.339884 | 1999 | 29986 | 3087.712193 |
11 | 3000 | 498 | 7457 | 98.32743 | 499 | 7472 | 93.101397 |
11 | 6000 | 498 | 7457 | 385.026068 | 499 | 7472 | 367.787958 |
11 | 9000 | 498 | 7457 | 794.07629 | 498 | 7457 | 774.825767 |
12 | 3000 | 1999 | 2000 | 51.059127 | 1999 | 2000 | 46.238696 |
12 | 6000 | 1999 | 2000 | 199.322478 | 1999 | 2000 | 185.71919 |
12 | 9000 | 1999 | 2000 | 405.680601 | 1999 | 2000 | 391.234908 |
13 | 3000 | 1 | 2 | 0.0312 | 1 | 2 | 0.0624 |
13 | 6000 | 1 | 2 | 0.156001 | 1 | 2 | 0.187201 |
13 | 9000 | 1 | 2 | 0.140401 | 1 | 2 | 0.249602 |
14 | 3000 | 1999 | 29972 | 400.220565 | 1999 | 29973 | 362.671125 |
14 | 6000 | 1999 | 29972 | 1544.316299 | 1999 | 29973 | 1460.294161 |
14 | 9000 | 1999 | 29972 | 3197.287295 | 1999 | 29973 | 3105.168705 |
15 | 3000 | 4 | 61 | 0.733205 | 4 | 61 | 0.733205 |
15 | 6000 | 4 | 61 | 3.790824 | 4 | 61 | 3.026419 |
15 | 9000 | 4 | 61 | 6.552042 | 4 | 61 | 6.146439 |
16 | 3000 | 5 | 62 | 1.060807 | 5 | 62 | 0.858006 |
16 | 6000 | 5 | 62 | 3.400822 | 5 | 62 | 3.291621 |
16 | 9000 | 5 | 62 | 6.942044 | 5 | 62 | 6.25564 |
17 | 3000 | 6 | 77 | 1.326009 | 6 | 91 | 1.216808 |
17 | 6000 | 6 | 77 | 4.243227 | 6 | 91 | 4.570829 |
17 | 9000 | 6 | 77 | 8.548855 | 6 | 91 | 9.40686 |
18 | 3000 | 5 | 76 | 0.936006 | 5 | 76 | 0.920406 |
18 | 6000 | 5 | 76 | 3.900025 | 5 | 76 | 3.775224 |
18 | 9000 | 5 | 76 | 8.533255 | 5 | 76 | 7.86245 |
19 | 3000 | 108 | 1060 | 15.5689 | 141 | 1272 | 17.565713 |
19 | 6000 | 81 | 788 | 44.429085 | 114 | 1029 | 53.820345 |
19 | 9000 | 63 | 628 | 70.512452 | 100 | 903 | 99.715839 |