NUMERICAL EXPERIMENTS

4.1 Example 1

Let us start from this problem defined by (Liu and Atluri, 2008a) i.e.

Minimize x² subject to x ≥ 1

After we did manipulate algebra at above problem its Lagrangian is

f (x) = x², g(x) = x − 1,

L = x²− λ(x − 1).

By using the KKT conditions and NCP, we get

Q1 : 2x − λ = 0,

H_k : λ ≥ 0, x − 1 ≥ 0, λ(x − 1) = 0 ⇐⇒ φ_k(λ, x − 1) = 0.

CHAPTER 4 NUMERICAL EXPERIMENTS

The ordinary differential equations (ODEs) are

Q₁ = − z1

CHAPTER 4 NUMERICAL EXPERIMENTS

Figure 4.1: Performance profile of φ^p_k when p = 50, x⁰_j = 1e − 8, and  = 10⁻⁵ in n dimensions

CHAPTER 4 NUMERICAL EXPERIMENTS

Figure 4.2: Performance profile of φ^p_kwhen p = 100, x⁰_j = 1e−5,  = 10⁻⁶ in n dimensions

CHAPTER 4 NUMERICAL EXPERIMENTS

Figure 4.3: Performance profile of φ^p_k when p > 1, x⁰_j = 1e − 2, and  = 10⁻⁴ in n dimensions

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.1: The solution of φ1 and φ2 in 2 dimensions by p = 2, 4, 10

p = 2 p = 4 p = 10

time φ₁ φ₂ time φ₁ φ₂ time φ₁ φ₂

0.0000 1.5000 2.0000 0.0000 1.5000 2.0000 0.0000 1.5000 2.0000 0.5804 1.5616 2.0756 0.4243 1.5643 2.0702 0.3817 1.5645 2.0650 1.1607 1.6044 2.1266 0.8486 1.6124 2.1215 0.7633 1.6129 2.1136 1.7411 1.6376 2.1656 1.2729 1.6511 2.1621 1.1450 1.6518 2.1526 2.3215 1.6654 2.1980 1.6973 1.6837 2.1962 1.5267 1.6847 2.1855 3.2935 1.7029 2.2412 2.5000 1.7337 2.2479 2.2676 1.7362 2.2370 4.2655 1.7331 2.2756 3.3027 1.7734 2.2888 3.0085 1.7770 2.2778 5.2375 1.7584 2.3043 4.1054 1.8064 2.3227 3.7493 1.8108 2.3116 6.2096 1.7803 2.3290 4.9081 1.8348 2.3518 4.4902 1.8399 2.3407 8.5977 1.8242 2.3782 6.7637 1.8883 2.4063 6.1733 1.8935 2.3943 10.9859 1.8585 2.4163 8.6193 1.9302 2.4488 7.8564 1.9356 2.4364 13.3740 1.8869 2.4475 10.4749 1.9647 2.4838 9.5394 1.9702 2.4711 15.7622 1.9111 2.4742 12.3305 1.9942 2.5137 11.2225 1.9999 2.5008 18.2622 1.9332 2.4983 14.8305 2.0281 2.5479 13.7225 2.0372 2.5380 20.7622 1.9526 2.5195 17.3305 2.0570 2.5771 16.2225 2.0685 2.5694 23.2622 1.9700 2.5385 19.8305 2.0822 2.6026 18.7225 2.0956 2.5965 25.7622 1.9858 2.5556 22.3305 2.1046 2.6251 21.2225 2.1195 2.6204 28.2622 2.0002 2.5712 24.8305 2.1247 2.6454 23.7225 2.1408 2.6417 30.7622 2.0134 2.5856 27.3305 2.1430 2.6638 26.2225 2.1601 2.6609 33.2622 2.0257 2.5988 29.8305 2.1597 2.6807 28.7225 2.1777 2.6785 35.7622 2.0372 2.6112 32.3305 2.1752 2.6962 31.2225 2.1938 2.6947 38.2622 2.0479 2.6228 34.8305 2.1895 2.7107 33.7225 2.2088 2.7096 40.7622 2.0580 2.6336 37.3305 2.2029 2.7241 36.2225 2.2227 2.7235 43.2622 2.0675 2.6438 39.8305 2.2154 2.7367 38.7225 2.2357 2.7365 45.7622 2.0765 2.6535 42.3305 2.2272 2.7486 41.2225 2.2479 2.7487 48.2622 2.0850 2.6627 44.8305 2.2383 2.7597 43.7225 2.2594 2.7602 50.7622 2.0932 2.6714 47.3305 2.2488 2.7703 46.2225 2.2703 2.7711 53.2622 2.1009 2.6797 49.8305 2.2588 2.7804 48.7225 2.2806 2.7814 55.7622 2.1084 2.6877 52.3305 2.2684 2.7900 51.2225 2.2904 2.7912 58.2622 2.1155 2.6953 54.8305 2.2775 2.7991 53.7225 2.2997 2.8006 60.7622 2.1223 2.7026 57.3305 2.2862 2.8078 56.2225 2.3087 2.8095 63.2622 2.1289 2.7096 59.8305 2.2945 2.8162 58.7225 2.3172 2.8181 65.7622 2.1352 2.7164 62.3305 2.3025 2.8242 61.2225 2.3254 2.8263 68.2622 2.1413 2.7229 64.8305 2.3102 2.8320 63.7225 2.3333 2.8342 70.7622 2.1472 2.7292 67.3305 2.3176 2.8394 66.2225 2.3409 2.8417

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.2: The solution of φ_k in 5 dimensions by p = 2 p = 2

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.0000 5.0000 3.0000 2.5000 0.5804 1.5606 2.0756 5.0884 3.0739 2.5710 1.1607 1.6024 2.1266 5.1473 3.1200 2.6254 1.7411 1.6316 2.1656 5.1919 3.1626 2.6667 2.3215 1.6624 2.1980 5.2288 3.2079 2.6910 3.2935 1.7009 2.2412 5.2779 3.2448 2.7466 4.2655 1.7311 2.2756 5.3168 3.2820 2.7828 5.2375 1.7514 2.3043 5.3490 3.3229 2.8129 6.2096 1.7813 2.3290 5.3767 3.3495 2.8418 8.5977 1.8202 2.3782 5.4315 3.4021 2.8951 10.9859 1.8505 2.4163 5.4739 3.4428 2.9388 13.3740 1.8809 2.4475 5.5084 3.4761 2.9714 15.7622 1.9011 2.4742 5.5379 3.5045 3.0012 18.2622 1.9132 2.4983 5.5645 3.5391 3.0453 20.7622 1.9226 2.5195 5.5878 3.5696 3.1473 23.2622 1.9200 2.5385 5.6086 3.5977 3.1669 25.7622 1.9158 2.5556 5.6274 3.6208 3.1671 28.2622 2.0000 2.5712 5.6445 3.6473 3.1689 30.7622 2.0034 2.5856 5.6612 3.6624 3.1691 33.2622 2.0157 2.5988 5.6787 3.6884 3.1796 35.7622 2.0272 2.6112 5.6982 3.6997 3.1962 38.2622 2.0379 2.6228 5.7098 3.7127 3.2252 40.7622 2.0480 2.6336 5.7326 3.7238 3.2254 43.2622 2.0575 2.6438 5.7437 3.7366 3.2369 46.9622 2.0665 2.6535 5.7543 3.7489 3.2489 48.2622 2.0750 2.6627 5.7742 3.7609 3.2609 50.7622 2.0832 2.6714 5.7837 3.7721 3.2721 54.3622 2.0909 2.6797 5.7858 3.7824 3.2824 57.7622 2.1014 2.6877 5.7924 3.7932 3.2932 58.9622 2.1105 2.6953 5.8097 3.8012 3.3012 62.7622 2.1203 2.7026 5.8106 3.8102 3.3103 63.2622 2.1229 2.7096 5.8192 3.8184 3.3184 65.7622 2.1312 2.7164 5.8265 3.8265 3.3265

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.3: The solution of φ_k in 5 dimensions by p = 4 p = 4

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.0000 5.0000 3.0000 2.5000 0.4804 1.5616 2.0656 5.0784 3.0839 2.5810 1.0607 1.6044 2.1166 5.1273 3.1400 2.6354 1.6411 1.6376 2.1556 5.1719 3.1826 2.6767 2.3015 1.6654 2.1780 5.1988 3.2179 2.7110 3.2535 1.7029 2.2212 5.2479 3.2648 2.7566 4.2455 1.7331 2.2456 5.3068 3.3020 2.7928 5.2275 1.7584 2.3001 5.3290 3.3329 2.8229 6.2196 1.7803 2.3190 5.3567 3.3595 2.8488 8.5377 1.8242 2.3582 5.4015 3.4121 2.9001 10.9059 1.8585 2.4063 5.4439 3.4528 2.9399 13.3140 1.8869 2.4275 5.4884 3.4861 2.9724 15.7222 1.9111 2.4342 5.5179 3.5145 3.0032 18.2322 1.9332 2.4683 5.5445 3.5401 3.0553 20.7222 1.9526 2.5095 5.5778 3.5726 3.1573 23.2222 1.9700 2.5285 5.5986 3.5987 3.1679 25.7122 1.9858 2.5446 5.6214 3.6209 3.1687 28.2222 2.0002 2.5612 5.6425 3.6483 3.1690 30.7522 2.0134 2.5756 5.6611 3.6634 3.1692 33.2422 2.0257 2.5888 5.6786 3.6864 3.1794 35.7222 2.0372 2.6012 5.6952 3.6995 3.1952 38.2022 2.0479 2.6128 5.7097 3.7117 3.2242 40.7122 2.0580 2.6226 5.7246 3.7241 3.2244 43.2222 2.0675 2.6328 5.7387 3.7368 3.2370 46.7322 2.0765 2.6425 5.7493 3.7490 3.2490 48.2122 2.0850 2.6517 5.7642 3.7612 3.2612 50.7322 2.0932 2.6614 5.7737 3.7724 3.2724 54.2222 2.1009 2.6697 5.7838 3.7826 3.2826 56.7222 2.1084 2.6772 5.7914 3.7934 3.2934 58.8122 2.1155 2.6852 5.8077 3.8014 3.3014 61.7322 2.1223 2.7016 5.8101 3.8106 3.3106 63.8222 2.1289 2.7086 5.8186 3.8186 3.3186 65.7122 2.1352 2.7144 5.8264 3.8268 3.3268

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.4: The solution of φ_k in 5 dimensions by p = 10 p = 10

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.0000 5.0000 3.0000 2.5000 0.3817 1.5645 2.0850 5.0650 3.0650 2.5650 0.7633 1.6129 2.1336 5.1136 3.1136 2.6136 1.1450 1.6518 2.1676 5.1526 3.1526 2.6526 1.5267 1.6847 2.1965 5.1855 3.1855 2.6855 2.2676 1.7362 2.2470 5.2370 3.2370 2.7370 3.0085 1.7770 2.2778 5.2778 3.2778 2.7778 3.0493 1.8108 2.3116 5.3116 3.3116 2.8116 4.4902 1.8399 2.3407 5.3407 3.3407 2.8407 5.1733 1.8935 2.3943 5.3943 3.3943 2.8943 6.8564 1.9356 2.4364 5.4364 3.4364 2.9364 8.5394 1.9702 2.4711 5.4711 3.4711 2.9711 10.2225 1.9999 2.5008 5.5008 3.5008 3.0008 12.7225 2.0372 2.5380 5.5381 3.5381 3.0381 14.2225 2.0685 2.5694 5.5694 3.5694 3.0694 17.7225 2.0956 2.5965 5.5965 3.5965 3.0965 19.2225 2.1195 2.6204 5.6204 3.6204 3.1204 22.7225 2.1408 2.6417 5.6417 3.6417 3.1417 24.2225 2.1601 2.6609 5.6610 3.6610 3.1610 28.1125 2.1777 2.6785 5.6785 3.6785 3.1785 30.1115 2.1938 2.6947 5.6947 3.6947 3.1947 33.1225 2.2088 2.7096 5.7096 3.7096 3.2096 34.2225 2.2227 2.7235 5.7235 3.7235 3.2235 38.0025 2.2357 2.7365 5.7365 3.7365 3.2365 40.2225 2.2479 2.7487 5.7487 3.7487 3.2487 43.2015 2.2594 2.7602 5.7602 3.7602 3.2602 46.2225 2.2703 2.7711 5.7711 3.7711 3.2711 47.7225 2.2806 2.7814 5.7814 3.7814 3.2814 50.2225 2.2904 2.7912 5.7912 3.7912 3.2912 53.7225 2.2997 2.8006 5.8006 3.8006 3.3006 56.2225 2.3087 2.8095 5.8095 3.8095 3.3095 58.7225 2.3172 2.8181 5.8181 3.8181 3.3181 61.2225 2.3254 2.8263 5.8263 3.8263 3.3263

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.5: The solution of φ_k in 6 dimensions by p = 2 p = 2

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.0000 5.0000 3.0000 2.5000 0.5804 1.5616 2.0656 5.0884 3.0639 2.5610 1.1607 1.6044 2.1166 5.1473 3.1200 2.6154 1.7411 1.6376 2.1556 5.1919 3.1626 2.6567 2.3215 1.6654 2.1880 5.2288 3.1879 2.6910 3.2935 1.7029 2.2412 5.2779 3.2501 2.7466 4.2655 1.7331 2.2756 5.3168 3.2900 2.7828 5.2375 1.7584 2.3043 5.3490 3.3229 2.8129 6.2096 1.7803 2.3290 5.3767 3.3515 2.8488 8.5977 1.8242 2.3782 5.4315 3.4021 2.9001 10.9859 1.8585 2.4163 5.4739 3.4428 2.9399 13.3740 1.8869 2.4475 5.5084 3.4861 2.9724 15.7622 1.9111 2.4742 5.5379 3.5145 3.0002 18.2622 1.9332 2.4983 5.5645 3.5401 3.0253 20.7622 1.9526 2.5195 5.5878 3.5626 3.0473 23.2622 1.9700 2.5385 5.6086 3.5827 3.0669 25.7622 1.9858 2.5556 5.6374 3.6008 3.0847 28.2622 2.0002 2.5712 5.6545 3.6173 3.1009 30.7622 2.0134 2.5856 5.6702 3.6324 3.1157 33.2622 2.0257 2.5988 5.6867 3.6464 3.1294 35.7622 2.0372 2.6112 5.7082 3.6595 3.1422 38.2622 2.0479 2.6228 5.7168 3.6717 3.1542 40.7622 2.0580 2.6336 5.7296 3.6831 3.1654 43.2622 2.0675 2.6438 5.7437 3.6938 3.1759 45.7622 2.0765 2.6535 5.7543 3.7040 3.1859 48.2622 2.0850 2.6627 5.7662 3.7137 3.1954 50.7622 2.0932 2.6714 5.7767 3.7229 3.2044 53.2622 2.1009 2.6797 5.7868 3.7316 3.2130 55.7622 2.1084 2.6877 5.7964 3.7400 3.2212 58.2622 2.1155 2.6953 5.8097 3.7479 3.2291 60.7622 2.1223 2.7026 5.8176 3.7556 3.2366 63.2622 2.1289 2.7096 5.8252 3.7630 3.2438 65.7622 2.1352 2.7164 5.8325 3.7701 3.2508

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.6: The solution of φ_k in 6 dimensions by p = 4 p = 4

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.0000 5.0000 3.0000 2.5000 0.4243 1.5623 2.0702 5.0712 3.0711 2.5709 0.8486 1.6114 2.1215 5.1231 3.1229 2.6226 1.2729 1.6411 2.1621 5.1641 3.1639 2.6635 1.6973 1.6737 2.1962 5.1985 3.1982 2.6978 2.5000 1.7137 2.2479 5.2508 3.2504 2.7499 3.3027 1.7534 2.2888 5.2920 3.2916 2.7910 4.1054 1.7764 2.3227 5.3261 3.3256 2.8250 4.9081 1.8048 2.3518 5.3553 3.3549 2.8542 6.7637 1.8383 2.4063 5.4101 3.4096 2.9089 8.6193 1.9002 2.4488 5.4529 3.4523 2.9515 10.4749 1.9147 2.4838 5.4880 3.4874 2.9866 12.3305 1.9242 2.5137 5.5181 3.5175 3.0166 14.8305 2.0081 2.5479 5.5525 3.5518 3.0509 17.3305 2.0170 2.5771 5.5818 3.5811 3.0802 19.8305 2.0222 2.6026 5.6073 3.6066 3.1057 22.3305 2.0946 2.6251 5.6300 3.6293 3.1284 24.8305 2.1047 2.6454 5.6504 3.6496 3.1487 27.3305 2.1130 2.6638 5.6688 3.6681 3.1671 29.8305 2.1297 2.6807 5.6858 3.6850 3.1840 32.3305 2.1352 2.6962 5.7013 3.7006 3.1996 34.8305 2.1495 2.7107 5.7158 3.7150 3.2140 37.3305 2.1529 2.7241 5.7293 3.7285 3.2275 39.8305 2.1754 2.7367 5.7419 3.7411 3.2401 42.3305 2.2072 2.7486 5.7538 3.7530 3.2520 44.8305 2.2183 2.7597 5.7650 3.7642 3.2632 47.3305 2.2288 2.7703 5.7757 3.7748 3.2738 49.8305 2.2388 2.7804 5.7857 3.7849 3.2839 52.3305 2.2484 2.7900 5.7953 3.7945 3.2935 54.8305 2.2575 2.7991 5.8045 3.8037 3.3026 57.3305 2.2662 2.8078 5.8133 3.8124 3.3114 59.8305 2.2745 2.8162 5.8217 3.8208 3.3197 62.3305 2.2825 2.8242 5.8297 3.8289 3.3278

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.7: The solution of φ_k in 6 dimensions by p = 10 p = 10

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.0000 5.0000 3.0000 2.5000 0.4143 1.5643 2.0802 5.0612 3.0811 2.5809 0.8386 1.6124 2.1315 5.1131 3.1329 2.6326 1.2529 1.6511 2.1721 5.1541 3.1739 2.6735 1.6773 1.6837 2.1982 5.1785 3.2082 2.7078 2.4000 1.7337 2.2579 5.2308 3.2604 2.7599 3.2027 1.7734 2.2988 5.2720 3.2936 2.7980 4.0054 1.8064 2.3427 5.3161 3.3286 2.8281 4.7081 1.8348 2.3528 5.3453 3.3569 2.8562 6.5637 1.8883 2.4163 5.4001 3.4196 2.9099 8.4193 1.9302 2.4588 5.4329 3.4583 2.9545 10.2749 1.9647 2.4848 5.4480 3.4894 2.9876 12.1305 1.9942 2.5237 5.5081 3.5185 3.0186 14.2305 2.0281 2.5579 5.5225 3.5538 3.0809 17.2305 2.0570 2.5871 5.5318 3.5841 3.0902 19.1305 2.0822 2.6126 5.5873 3.6086 3.1157 22.1305 2.1046 2.6351 5.6100 3.6393 3.1384 24.4305 2.1247 2.6554 5.6204 3.6596 3.1587 27.2305 2.1430 2.6738 5.6388 3.6781 3.1871 29.4305 2.1597 2.6907 5.6458 3.6950 3.1960 32.1305 2.1752 2.6982 5.6713 3.7106 3.2096 34.3305 2.1895 2.7117 5.7058 3.7250 3.2160 37.1305 2.2029 2.7261 5.7193 3.7385 3.2375 39.5305 2.2154 2.7377 5.7219 3.7521 3.2461 42.2305 2.2272 2.7496 5.7338 3.7620 3.2580 44.4305 2.2383 2.7697 5.7450 3.7742 3.2652 47.1305 2.2488 2.7713 5.7557 3.7838 3.2758 49.2305 2.2588 2.7824 5.7657 3.7919 3.2869 52.1305 2.2684 2.7910 5.7753 3.8045 3.2975 54.5305 2.2775 2.7998 5.7945 3.8137 3.3056 57.1305 2.2862 2.8088 5.8033 3.8224 3.3184 59.4305 2.2945 2.8262 5.8117 3.8308 3.3297 62.1305 2.3025 2.8342 5.8197 3.8419 3.3378

CHAPTER4NUMERICALEXPERIMENTS Table 4.8: The solution of φ₃, φ₄, and φ₅ in 3 dimensions by p = 2, 4, 10

p = 2 p = 4 p = 10

time φ₃ φ₄ φ₅ time φ₃ φ₄ φ₅ time φ₃ φ₄ φ₅

0.0000 1.5000 2.0000 5.0000 0.0000 1.5000 2.0000 5.0000 0.0000 1.5000 2.0000 5.0000 0.5804 1.5616 2.0756 5.0884 0.4243 1.5643 2.0702 5.0712 0.3817 1.5645 2.0650 5.0650 1.1607 1.6044 2.1266 5.1473 0.8486 1.6124 2.1215 5.1231 0.7633 1.6129 2.1136 5.1136 1.7411 1.6376 2.1656 5.1919 1.2729 1.6511 2.1621 5.1641 1.1450 1.6518 2.1526 5.1526 2.3215 1.6654 2.2980 5.2288 1.6973 1.6837 2.1962 5.1985 1.5267 1.6847 2.1855 5.1855 3.2935 1.7029 2.3412 5.2779 2.5000 1.7337 2.2479 5.2508 2.2676 1.7362 2.2370 5.2370 4.2655 1.7331 2.3756 5.3168 3.3027 1.7734 2.2888 5.2920 3.0085 1.7770 2.2778 5.2778 5.2375 1.7584 2.4043 5.3490 4.1054 1.8064 2.3227 5.3261 3.7493 1.8108 2.3116 5.3116 6.2096 1.7803 2.4290 5.3767 4.9081 1.8348 2.3518 5.3553 4.4902 1.8399 2.3407 5.3407 8.5977 1.8242 2.4782 5.4315 6.7637 1.8883 2.4063 5.4101 6.1733 1.8935 2.3943 5.3943 10.9859 1.8585 2.5163 5.4739 8.6193 1.9302 2.4488 5.4529 7.8564 1.9356 2.4364 5.4364 13.3740 1.8869 2.5475 5.5084 10.4749 1.9647 2.4838 5.4880 9.5394 1.9702 2.4711 5.4711 15.7622 1.9111 2.5742 5.5379 12.3305 1.9942 2.5137 5.5181 11.2225 1.9999 2.5008 5.5008 18.2622 1.9332 2.5983 5.5645 14.8305 2.0281 2.5479 5.5525 13.7225 2.0372 2.5380 5.5381 20.7622 1.9526 2.6195 5.5878 17.3305 2.0570 2.5771 5.5818 16.2225 2.0685 2.5694 5.5694 23.2622 1.9700 2.6385 5.6086 19.8305 2.0822 2.6026 5.6073 18.7225 2.0956 2.5965 5.5965 25.7622 1.9858 2.6556 5.6274 22.3305 2.1046 2.6251 5.6230 21.2225 2.1195 2.6204 5.6204 28.2622 2.0002 2.6712 5.6445 24.8305 2.1247 2.6454 5.6434 23.7225 2.1408 2.6417 5.6417 30.7622 2.0134 2.6856 5.6692 27.3305 2.1430 2.6638 5.6688 26.2225 2.1601 2.6609 5.6610 33.2622 2.0257 2.6988 5.6847 29.8305 2.1597 2.6797 5.6858 28.7225 2.1777 2.6785 5.6785 35.7622 2.0372 2.7112 5.7082 32.3305 2.1752 2.6962 5.7013 31.2225 2.1938 2.6947 5.6947 38.2622 2.0479 2.7228 5.7118 34.8305 2.1895 2.7107 5.7158 33.7225 2.2088 2.7096 5.7096 40.7622 2.0580 2.7336 5.7286 37.3305 2.2029 2.7241 5.7293 36.2225 2.2227 2.7235 5.7235 43.2622 2.0675 2.7438 5.7387 39.8305 2.2154 2.7367 5.7419 38.7225 2.2357 2.7365 5.7365 45.7622 2.0765 2.7535 5.7493 42.3305 2.2272 2.7489 5.7538 41.2225 2.2479 2.7487 5.7487 48.2622 2.0850 2.7798 5.7801 44.8305 2.2383 2.7797 5.7650 43.7225 2.2594 2.7602 5.7602 50.7622 2.0932 2.7744 5.7937 47.3305 2.2488 2.7811 5.7757 46.2225 2.2703 2.7711 5.7711

41

CHAPTER 4 NUMERICAL EXPERIMENTS

4.2 Example 2

Suppose that the multivariate optimization problem with inequality constraints is Minimize − xe^−x2−y2

subject to x, y ≥ 0 Its Lagrangian is

L = −xe^−x2−y2 − λx − λy.

By using the same ways with previous example

Q₁(x) : ∂(−xe^−x2−y2 − λx − λy)

∂x = 0,

Q₁(y) : ∂(−xe^−x2−y2 − λx − λy)

∂y = 0,

H_k(x) : λ ≥ 0, x ≥ 0, λx = 0 ⇐⇒ φ_k(x)(λ, x) = 0, Hk(y) : λ ≥ 0, y ≥ 0, λy = 0 ⇐⇒ φk(y)(λ, y) = 0.

Such the ODEs that is

H_k(x) = − z₃

1 + τφ^p_k(x)(λ, x), Hk(y) = − z₃

1 + τφ^p_k(y)(λ, y).

CHAPTER 4 NUMERICAL EXPERIMENTS

Figure 4.4: The graph of f (x, y) = −xe^−x2−y2 for x = −2 : 0.2 : 2 and y = −2 : 0.2 : 2

Figure 4.5: Performance profile of H_k(x) and H_k(y) in n dimensions solved by FTIM and they depend on φ^p_k by p > 1, x⁰_j = 1e − 4,  = 10⁻³

CHAPTER 4 NUMERICAL EXPERIMENTS

Figure 4.6: Performance profile of H_k in n dimensions solved by FTIM and every single p compared where x⁰_j = 1e − 4,  = 10⁻³

Figure 4.7: Performance profile of Hk in n dimensions solved by FTIM where x⁰_j is different while p = 4 and  = 10⁻³

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.9: The solution of φ²_k in 9 dimensions [H_k(x)]

p = 2

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.1000 3.4000 4.1000 5.2000 0.4145 1.5650 2.1811 3.4898 4.1916 5.2932 0.8291 1.6147 2.2412 3.5558 4.2588 5.3615 1.2436 1.6552 2.2893 3.6082 4.3121 5.4157 1.6582 1.6897 2.3298 3.6523 4.3570 5.4613 2.4457 1.7435 2.3922 3.7198 4.4256 5.5309 3.2331 1.7870 2.4417 3.7731 4.4798 5.5859 4.0206 1.8234 2.4830 3.8174 4.5247 5.6315 4.8081 1.8550 2.5185 3.8554 4.5633 5.6706 6.6310 1.9152 2.5855 3.9268 4.6357 5.7440 8.4539 1.9628 2.6380 3.9826 4.6923 5.8014 10.2769 2.0024 2.6814 4.0286 4.7388 5.8485 12.0998 2.0365 2.7185 4.0679 4.7787 5.8889 14.5998 2.0765 2.7619 4.1137 4.8250 5.9358 17.0998 2.1107 2.7989 4.1526 4.8645 5.9757

Table 4.10: The solution of φ⁴_k in 9 dimensions [H_k(x)]

p = 4

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.1000 3.4000 4.1000 5.2000 0.3031 1.5672 2.1735 3.4743 4.1743 5.2743 0.6062 1.6215 2.2314 3.5326 4.2327 5.3327 0.9092 1.6671 2.2793 3.5808 4.2809 5.3810 1.2123 1.7064 2.3203 3.6221 4.3223 5.4223 1.8760 1.7772 2.3935 3.6958 4.3959 5.4960 2.5396 1.8335 2.4512 3.7537 4.4539 5.5540 3.2033 1.8803 2.4990 3.8017 4.5019 5.6020 3.8669 1.9205 2.5399 3.8428 4.5430 5.6431 5.3165 1.9924 2.6128 3.9159 4.6162 5.7163 6.7661 2.0493 2.6703 3.9736 4.6739 5.7740 8.2156 2.0965 2.7180 4.0214 4.7216 5.8218 9.6652 2.1370 2.7588 4.0623 4.7626 5.8628 12.1652 2.1954 2.8177 4.1214 4.8216 5.9218 14.6652 2.2436 2.8662 4.1700 4.8703 5.9704

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.11: The solution of φ¹⁰_k in 9 dimensions [H_k(x)]

p = 10

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.1000 3.4000 4.1000 5.2000 0.2726 1.5671 2.1677 3.4677 4.1677 5.2677 0.5452 1.6212 2.2219 3.5219 4.2219 5.3219 0.8179 1.6666 2.2673 3.5673 4.2673 5.3673 1.0905 1.7057 2.3065 3.6065 4.3065 5.4065 1.7177 1.7794 2.3802 3.6802 4.3802 5.4802 2.3450 1.8374 2.4382 3.7382 4.4382 5.5382 2.9722 1.8854 2.4862 3.7862 4.4862 5.5862 3.5995 1.9265 2.5273 3.8273 4.5273 5.6273 4.9244 1.9976 2.5984 3.8984 4.5984 5.6984 6.2492 2.0540 2.6548 3.9548 4.6548 5.7548 7.5741 2.1008 2.7017 4.0017 4.7017 5.8017 8.8989 2.1411 2.7420 4.0420 4.7420 5.8420 11.3989 2.2043 2.8051 4.1051 4.8051 5.9051 13.8989 2.2557 2.8565 4.1565 4.8565 5.9565

Table 4.12: The solution of φ²_k in 11 dimensions [H_k(y)]

p = 2

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 1.1000 1.3000 1.2000 1.4000 0.4145 1.5650 1.1234 1.3510 1.2397 1.4591 0.8291 1.6147 1.1434 1.3915 1.2722 1.5049 1.2436 1.6552 1.1612 1.4252 1.2999 1.5426 1.6582 1.6897 1.1772 1.4544 1.3242 1.5750 2.4608 1.7445 1.2044 1.5015 1.3642 1.6266 3.2635 1.7885 1.2280 1.5400 1.3976 1.6683 4.0661 1.8254 1.2488 1.5727 1.4262 1.7035 4.8688 1.8573 1.2676 1.6012 1.4515 1.7340 6.7052 1.9173 1.3046 1.6554 1.5000 1.7917 8.5416 1.9649 1.3357 1.6988 1.5394 1.8375 10.3781 2.0044 1.3625 1.7352 1.5728 1.8758 12.2145 2.0385 1.3862 1.7666 1.6018 1.9088 14.7145 2.0782 1.4147 1.8035 1.6360 1.9473 17.2145 2.1122 1.4397 1.8353 1.6656 1.9804

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.13: The solution of φ⁴_k in 11 dimensions [H_k(y)]

p = 4

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 1.1000 1.3000 1.2000 1.4000 0.3031 1.5672 1.1203 1.3533 1.2390 1.4622 0.6062 1.6215 1.1390 1.3992 1.2741 1.5137 0.9092 1.6671 1.1567 1.4393 1.3061 1.5575 1.2123 1.7064 1.1734 1.4748 1.3355 1.5956 1.8894 1.7785 1.2080 1.5416 1.3930 1.6660 2.5664 1.8356 1.2395 1.5957 1.4418 1.7222 3.2435 1.8829 1.2685 1.6412 1.4838 1.7689 3.9205 1.9236 1.2954 1.6805 1.5207 1.8091 5.3809 1.9952 1.3471 1.7504 1.5873 1.8802 6.8412 2.0520 1.3919 1.8061 1.6411 1.9365 8.3016 2.0990 1.4312 1.8525 1.6864 1.9834 9.7620 2.1395 1.4661 1.8924 1.7255 2.0236 12.2620 2.1975 1.5177 1.9498 1.7819 2.0814 14.7620 2.2453 1.5616 1.9973 1.8288 2.1292

Table 4.14: The solution of φ¹⁰_k in 11 dimensions [H_k(y)]

p = 10

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 1.1000 1.3000 1.2000 1.4000 0.2706 1.5666 1.1183 1.3539 1.2365 1.4640 0.5412 1.6205 1.1354 1.4024 1.2707 1.5173 0.8118 1.6656 1.1516 1.4454 1.3029 1.5623 1.0824 1.7046 1.1671 1.4834 1.3335 1.6011 1.6938 1.7769 1.2001 1.5549 1.3965 1.6733 2.3051 1.8340 1.2309 1.6119 1.4510 1.7304 2.9165 1.8814 1.2600 1.6593 1.4977 1.7778 3.5278 1.9221 1.2877 1.6999 1.5378 1.8184 4.8296 1.9930 1.3427 1.7708 1.6085 1.8894 6.1314 2.0494 1.3924 1.8271 1.6647 1.9457 7.4333 2.0962 1.4366 1.8739 1.7115 1.9925 8.7351 2.1365 1.4756 1.9142 1.7518 2.0328 11.2351 2.2006 1.5389 1.9783 1.8159 2.0969 13.7351 2.2526 1.5907 2.0303 1.8678 2.1489

CHAPTER 4 NUMERICAL EXPERIMENTS

4.3 Example 3

Let us define the multivariate optimization problem with inequality constraints that is Minimize cos xy

subject to x + a, y + a ≥ 0, a ∈ R⁺ Its Lagrangian is

L = cos xy − λ(x + a) − λ(y + a).

By using the same ways with previous example

Q₁(x) : − y sin xy − λ = 0, Q₁(y) : − x sin xy − λ = 0,

H_k(x) : λ ≥ 0, (x + a) ≥ 0, λ(x + a) = 0 ⇐⇒ φ_k(x)(λ, x + a) = 0, Hk(y) : λ ≥ 0, (y + a) ≥ 0, λ(y + a) = 0 ⇐⇒ φk(y)(λ, y + a) = 0.

Such the ODEs that is

Q_k(x) = − z₁

1 + τ(−y sin xy − λ), Q_k(y) = − z₁

1 + τ(−x sin xy − λ), H_k(x) = − z₃

1 + τφ^p_k(x)(λ, x + a), H_k(y) = − z₃

1 + τφ^p_k(y)(λ, y + a).

CHAPTER 4 NUMERICAL EXPERIMENTS

Figure 4.8: The graph of f (x, y) = cos xy for x = [−π, π] and y = [−π, π]

Figure 4.9: Performance profile of φ^p_k in n dimensions solved by FTIM where p >

1 and  = 10⁻⁷

CHAPTER 4 NUMERICAL EXPERIMENTS

Figure 4.10: Performance profile of φ^p_k in n dimensions solved by FTIM wherever x⁰_j = 1e − 6 and  = 10⁻⁹

Figure 4.11: Performance profile of φ^p_kin n dimensions solved by FTIM when p > 1, x⁰_j = 1e − 4, and  = 10⁻⁵

Figure 4.12: Performance profile of Qkand Hkin n dimensions solved by several numerical methods when p > 1, x⁰_j = 1e − 2, and  = 10⁻³

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.15: The solution of φ²_k in 17 dimensions [H_k(x)]

p = 2

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.1000 3.4000 4.1000 5.2000 0.3675 1.5664 2.1870 3.4990 4.2014 5.3037 0.7351 1.6191 2.2537 3.5739 4.2781 5.3819 1.1026 1.6631 2.3081 3.6344 4.3400 5.4450 1.4702 1.7011 2.3544 3.6858 4.3924 5.4985 2.2005 1.7639 2.4294 3.7684 4.4767 5.5844 2.9308 1.8149 2.4893 3.8338 4.5434 5.6523 3.6612 1.8580 2.5392 3.8882 4.5987 5.7086 4.3915 1.8955 2.5822 3.9349 4.6463 5.7570 6.0538 1.9660 2.6624 4.0213 4.7343 5.8464 7.7162 2.0224 2.7256 4.0892 4.8033 5.9166 9.3785 2.0695 2.7780 4.1453 4.8602 5.9744 11.0408 2.1103 2.8230 4.1933 4.9090 6.0239 13.5408 2.1624 2.8803 4.2543 4.9709 6.0868 16.0408 2.2067 2.9287 4.3056 5.0230 6.1395

Table 4.16: The solution of φ⁴_k in 17 dimensions [H_k(x)]

p = 4

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.1000 3.4000 4.1000 5.2000 0.2655 1.5689 2.1809 3.4824 4.1825 5.2826 0.5311 1.6272 2.2463 3.5489 4.2490 5.3491 0.7966 1.6775 2.3013 3.6046 4.3048 5.4049 1.0622 1.7218 2.3490 3.6529 4.3531 5.4532 1.6751 1.8070 2.4393 3.7440 4.4444 5.5446 2.2879 1.8755 2.5106 3.8160 4.5164 5.6166 2.9008 1.9327 2.5698 3.8756 4.5760 5.6762 3.5137 1.9821 2.6206 3.9267 4.6271 5.7274 4.8459 2.0703 2.7107 4.0173 4.7179 5.8181 6.1782 2.1404 2.7820 4.0890 4.7895 5.8898 7.5104 2.1986 2.8411 4.1483 4.8489 5.9492 8.8427 2.2488 2.8919 4.1993 4.8999 6.0002 11.3427 2.3270 2.9709 4.2786 4.9792 6.0796 13.8427 2.3907 3.0351 4.3430 5.0436 6.1440

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.17: The solution of φ¹⁰_k in 17 dimensions [H_k(x)]

p = 10

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 2.1000 3.4000 4.1000 5.2000 0.2320 1.5695 2.1731 3.4731 4.1731 5.2731 0.4639 1.6288 2.2334 3.5334 4.2334 5.3334 0.6959 1.6799 2.2848 3.5848 4.2848 5.3848 0.9278 1.7245 2.3297 3.6297 4.3297 5.4297 1.5040 1.8161 2.4215 3.7216 4.4216 5.5216 2.0803 1.8884 2.4939 3.7939 4.4939 5.5939 2.6565 1.9482 2.5537 3.8537 4.5537 5.6537 3.2327 1.9995 2.6050 3.9051 4.6051 5.7051 4.4184 2.0861 2.6917 3.9917 4.6917 5.7917 5.6041 2.1553 2.7608 4.0608 4.7608 5.8608 6.7899 2.2129 2.8185 4.1185 4.8185 5.9185 7.9756 2.2626 2.8682 4.1682 4.8682 5.9682 10.4756 2.3488 2.9544 4.2544 4.9544 6.0544 12.9756 2.4177 3.0232 4.3232 5.0232 6.1232

Table 4.18: The solution of φ²_k in 25 dimensions [H_k(y)]

p = 2

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 1.1000 1.3000 1.2000 1.4000 0.3675 1.5664 1.1216 1.3501 1.2379 1.4594 0.7351 1.6191 1.1406 1.3914 1.2701 1.5072 1.1026 1.6631 1.1577 1.4267 1.2981 1.5476 1.4702 1.7011 1.1733 1.4578 1.3232 1.5826 2.2221 1.7656 1.2019 1.5114 1.3674 1.6426 2.9740 1.8176 1.2268 1.5558 1.4047 1.6915 3.7259 1.8615 1.2492 1.5937 1.4370 1.7329 4.4779 1.8996 1.2694 1.6269 1.4658 1.7690 6.1626 1.9701 1.3092 1.6893 1.5204 1.8361 7.8474 2.0264 1.3430 1.7398 1.5655 1.8901 9.5322 2.0735 1.3726 1.7825 1.6039 1.9354 11.2170 2.1143 1.3990 1.8196 1.6375 1.9746 13.7170 2.1658 1.4337 1.8669 1.6808 2.0244 16.2170 2.2096 1.4641 1.9074 1.7181 2.0668

CHAPTER 4 NUMERICAL EXPERIMENTS

Table 4.19: The solution of φ⁴_k in 25 dimensions [H_k(y)]

p = 4

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 1.1000 1.3000 1.2000 1.4000 0.2655 1.5689 1.1179 1.3500 1.2351 1.4614 0.5311 1.6272 1.1346 1.3950 1.2675 1.5147 0.7966 1.6775 1.1505 1.4358 1.2977 1.5617 1.0622 1.7218 1.1656 1.4729 1.3261 1.6035 1.6901 1.8089 1.1990 1.5491 1.3868 1.6870 2.3180 1.8785 1.2298 1.6126 1.4400 1.7546 2.9459 1.9366 1.2586 1.6668 1.4872 1.8115 3.5738 1.9866 1.2856 1.7140 1.5294 1.8606 4.9311 2.0752 1.3390 1.7989 1.6075 1.9480 6.2884 2.1456 1.3867 1.8672 1.6720 2.0177 7.6457 2.2041 1.4297 1.9244 1.7267 2.0757 9.0030 2.2543 1.4687 1.9737 1.7742 2.1257 11.5030 2.3314 1.5320 2.0496 1.8480 2.2023 14.0030 2.3944 1.5864 2.1117 1.9089 2.2650

Table 4.20: The solution of φ¹⁰_k in 25 dimensions [H_k(y)]

p = 10

time φ₁ φ₂ φ₃ φ₄ φ₅

0.0000 1.5000 1.1000 1.3000 1.2000 1.4000 0.2320 1.5695 1.1157 1.3471 1.2315 1.4613 0.4639 1.6288 1.1306 1.3913 1.2611 1.5166 0.6959 1.6799 1.1447 1.4330 1.2894 1.5659 0.9278 1.7245 1.1583 1.4721 1.3166 1.6098 1.4604 1.8099 1.1878 1.5519 1.3754 1.6944 1.9929 1.8783 1.2154 1.6189 1.4297 1.7627 2.5255 1.9354 1.2416 1.6758 1.4798 1.8198 3.0580 1.9847 1.2666 1.7248 1.5256 1.8691 4.1557 2.0687 1.3152 1.8085 1.6067 1.9530 5.2534 2.1361 1.3606 1.8759 1.6736 2.0205 6.3511 2.1926 1.4034 1.9324 1.7299 2.0769 7.4487 2.2414 1.4438 1.9812 1.7785 2.1257 9.7974 2.3275 1.5216 2.0672 1.8644 2.2118 12.1461 2.3962 1.5882 2.1360 1.9332 2.2805

Chapter 5 Conclusions

The FTIM gives the advantages a lot into numerical experiments and is giving an ap-proximation solution at nonlinear optimization problem better than others. The way to reformulate from NAEs into ODEs by means of fictitious time function until obtained a new numerical equation is a long way to be conducted, anyway this way gives us more satisfying results such as a higher stability, an approximate accuracy, and also efficient loop in algorithm when used the performance profile to find computing times and number of iterations. The comparisons of numerical methods in optimization problems together with the conjugate gradient, Newton, Broyden, and Levenberg-Marquardt, as an evi-dencing to reveal the performance of the fictitious time integration method (FTIM), is better than others.

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在文檔中 An ordinary differential equation approach for nonlinear programming and nonlinear complementarity problem (頁 30-59)