Path: utzoo!utgpu!news-server.csri.toronto.edu!bonnie.concordia.ca!uunet!mcsun!ukc!mucs!liv-cs!liv!ec13 From: EC13@LIVERPOOL.AC.UK Newsgroups: comp.theory Subject: Brand New Optimization Methods Message-ID: <91166.152726EC13@LIVERPOOL.AC.UK> Date: 15 Jun 91 14:27:26 GMT Organization: University of Liverpool Lines: 144 I am including here some numerical examples solved using the new entropy-based methods...... 1)In Vector Optimization to generate Pareto set: ------------------------------------------------ This example was taken from 'Multicriterion Optimization in Engineering', Osyczka, Ellis Horwood, Chichester, 1984. It is a beam design example. F1(X) = 0.785{x1 (6400 - x2**2) + (1000 - x1) (10000 - x2**2)} mm^3 ----> Min F2(X) = 3.298 x 10^(-5) {[(1/(4.096 x 10^6 - x2**4)) - (1/(10^8-x2**4))] * x1^3 + (10^9/(10^8 - x2**4))} mm/N ----> Min S.t.: g1(X) = 180 - 9.87 x 10^6 * x1 / (4.096*10^7 - x2**4) >=0 g2(X) = 75.2 - x2 >=0 g3(X) = x2 - 40 >=0 x1,x2>=o Using the 1st entropy-based method, which can generate convex solutions only, I was able to obtain 18 solutions while using the 2nd entropy-based method I was able to generate 53 solutions. The results are summarized below: ------------------------------------------------------------------------------- The Entropy-Based Weighted Objectives Function Method (EWOF) ------------------------------------------------------------------------------- No. X=(x1,x2) F(X)=(F1,F2) ------------------------------------------------------------------------------- 1 (165.29, 75.2) (0.2943695e+07, 0.49924662e-03) 2 (183.58, 74.609) (0.2961524e+07, 0.495371176e-03) 3 (192.75, 74.307) (0.2970895e+07, 0.493597705e-03) 4 (202.39, 73.986) (0.2981043e+07, 0.491856365e-03) 5 (212.51, 73.744) (0.2992045e+07, 0.490157399e-03) 6 (219.88, 73.392) (0.3000292e+07, 0.489001861e-03) 7 (223.67, 73.262) (0.3004618e+07, 0.488433288e-03) 8 (152.71, 40.0) (0.6162431e+07, 0.340317842e-03) 9 (145.21, 40.0) (0.6183632e+07, 0.340058003e-03) 10 (137.95, 40.0) (0.6204248e+07, 0.33983076e-03) 11 (132.72, 40.0) (0.6218924e+07, 0.33968105e-03) 12 (131.05, 40.0) (0.6223628e+07, 0.33963611e-03) 13 (124.50, 40.0) (0.624214 e+07, 0.339468941e-03) 14 (110.72, 40.0) (0.6281057e+07, 0.339171384e-03) 15 (101.02, 40.0) (0.6308531e+07, 0.339000951e-03) 16 (96.227, 40.0) (0.6321995e+07, 0.338929007e-03) 17 (74.569, 40.0) (0.6385737e+07, 0.33867266e-03) 18 (1.0, 40.0) (0.6591174e+07, 0.33846451e-03) ------------------------------------------------------------------------------- The Entropy-Based Constrained Objective Functions Method (ECOF) ------------------------------------------------------------------------------- 1 (165.30, 75.2) (0.2943734e+07, 0.4992401 e-03) 2 (182.30, 74.651) (0.2960245e+07, 0.4956268 e-03) 3 (194.01, 74.265) (0.2972194e+07, 0.4933646 e-03) 4 (206.16, 73.859) (0.2985099e+07, 0.4912079 e-03) 5 (217.11, 73.487) (0.2997181e+07, 0.4894268 e-03) 6 (221.51, 73.336) (0.3002174e+07, 0.4887516 e-03) 7 (221.43, 71.549) (0.3205612e+07, 0.466343 e-03) 8 (232.23, 71.191) (0.3215188e+07, 0.4652811 e-03) 9 (225.57, 67.174) (0.3670394e+07, 0.4277397 e-03) 10 (236.85, 66.252) (0.3735082e+07, 0.4232714 e-03) 11 (219.06, 66.058) (0.3805428e+07, 0.4189061 e-03) 12 (225.08, 65.402) (0.3856105e+07, 0.4156467 e-03) 13 (208.21, 64.240) (0.4022139e+07, 0.4063430 e-03) 14 (231.21, 62.351) (0.4144831e+07, 0.3994876 e-03) 15 (234.36, 62.070) (0.4163319e+07, 0.3985558 e-03) 16 (242.77, 61.416) (0.4203013e+07, 0.3966531 e-03) 17 (234.36, 58.967) (0.4458198e+07, 0.3850318 e-03) 18 (231.06, 58.955) (0.4468607e+07, 0.3845755 e-03) 19 (228.60, 58.361) (0.4530231e+07, 0.3820444 e-03) 20 (213.70, 57.052) (0.4691012e+07, 0.3758790 e-03) 21 (232.50, 55.351) (0.4787910e+07, 0.3725172 e-03) 22 (226.02, 54.971) (0.4839180e+07, 0.3707132 e-03 ) 23 (216.04, 54.552) (0.4903343e+07, 0.3685560 e-03) 24 (228.51, 53.746) (0.4936624e+07, 0.3675970 e-03) 25 (228.51, 52.481) (0.5042167e+07, 0.3644039 e-03) 26 (228.51, 51.345) (0.5134723e+07, 0.3617750 e-03) 27 (228.51, 50.906) (0.5169965e+07, 0.3608146 e-03) 28 (228.51, 50.241) (0.5222748e+07, 0.3594169 e-03) 29 (228.51, 49.383) (0.5289871e+07, 0.3577087 e-03) 30 (193.97, 48.748) (0.5436395e+07, 0.3538036 e-03) 31 (202.94, 47.265) (0.5522785e+07, 0.3518865 e-03 ) 32 (192.28, 46.483) (0.5610471e+07, 0.3499517 e-03) 33 (175.45, 46.399) (0.5664181e+07, 0.3488639 e-03) 34 (175.45, 44.620) (0.5791270e+07, 0.3463724 e-03) 35 (175.45, 42.870) (0.5911458e+07, 0.3442250 e-03) 36 (192.55, 40.0) (0.6049859e+07, 0.3421793 e-03) 37 (175.45, 40.0) (0.6098175e+07, 0.3412750 e-03) 38 (154.20, 40.0) (0.6158251e+07, 0.3403721 e-03) 39 (147.19, 40.0) (0.6178043e+07, 0.3401239 e-03) 40 (134.01, 40.0) (0.6215292e+07, 0.3397167 e-03 ) 41 (132.52, 40.0) (0.6219488e+07, 0.3396757 e-03) 42 (126.21, 40.0) (0.6237336e+07, 0.3395106 e-03) 43 (115.18, 40.0) (0.6268511e+07, 0.3392596 e-03) 44 (109.69, 40.0) (0.6284024e+07, 0.3391511 e-03) 45 (103.31, 40.0) (0.6302049e+07, 0.3390382 e-03) 46 (92.308, 40.0) (0.6333136e+07, 0.3388738 e-03) 47 (85.273, 40.0) (0.6353017e+07, 0.3387872 e-03) 48 (77.439, 40.0) (0.6375157e+07, 0.3387062 e-03) 49 (51.448, 40.0) (0.6448607e+07, 0.3385353 e-03 ) 50 (48.834, 40.0) (0.6455994e+07, 0.3385250 e-03) 51 (44.642, 40.0) (0.6467846e+07, 0.3385106 e-03) 52 (42.974, 40.0) (0.6472553e+07, 0.3385057 e-03) 53 (0.0, 40.0) (0.6593964e+07, 0.3384645 e-03) ------------------------------------------------------------------------------- 2)In Single-Criterion Minimization Using Simulated Entropy: ----------------------------------------------------------- This example was taken from 'Lecture Notes in Economic and Mathematical Systems', Hock W. and Schittokowski K., Springer-Verlag, 1981 Minimiza: F(X) = x1**2 + 0.5*x2**2 + x3**2 + 0.5*x4**2 - x1*x3 + x3*x4 - x1 - 3*x2 + x3 - x4 S.t.: - x1 - 2*x2 - x3 - x4 + 5 >= 0 - 3*x1 - x2 - 2*x3 + x4 + 4 >= 0 x2 + 4*x3 - 1.5 >= 0 x1,x2,x3,x4>=0 ------------------------------------------------------------------------------- Simulated Entropy ------------------------------------------------------------------------------- P F x1 x2 x3 x4 g1 g2 g3 Temp. Energy ------------------------------------------------------------------------------- 1.5 -2.8795 0.00011 1.4091 0.3661 0.299 -1.5165 -2.1574 -1.374 1.4 -3.162 0.0001 1.5283 0.3126 0.285 -1.3461 -2.1308 -1.279 1.3 -3.4176 0.041 1.5785 0.273 0.296 -1.2331 -2.0483 -1.171 1.2 -3.6759 0.10919 1.6194 0.2575 0.31 -1.0924 -1.84 -1.15 1.1 -3.9249 0.14549 1.6635 0.2059 0.314 -1.0079 -1.8019 -0.987 1.0 -4.1706 0.2857 1.7041 0.18542 0.320 -0.8776 -1.6195 -0.946 0.9 -4.4028 0.2168 1.74 0.15398 0.331 -0.7729 -1.4979 -0.856 0.8 *** -4.6366 0.31235 1.7787 0.11222 0.327 -0.691 -1.3868 -0.728 0.7 -4.8679 0.37211 1.8238 0.09149 0.333 -0.5556 -1.2102 -0.69 0.6 -5.0853 0.4255 1.8601 0.05629 0.333 -0.4654 -1.036 -0.585 0.5 -5.2948 0.48263 1.8925 0.02463 0.334 -0.3734 -0.9446 -0.491 0.4 -5.5005 0.5483 1.9251 0.0001 0.329 -0.272 -0.7591 -0.426 0.3 -5.6671 0.6199 1.9468 0.0001 0.336 -0.1506 -0.5291 -0.447 0.2 -5.7836 0.6751 1.9639 0.00086 0.331 -0.0651 -0.3401 -0.467 0.1 -5.8623 0.7306 1.962 0.0001 0.323 -0.0223 -0.1688 -0.462 0.075 -5.8662 0.7292 1.9635 0.0001 0.328 -0.0157 -0.1765 -0.464 0.05 -5.8867 0.73933 1.9814 0.0001 0.290 -0.0077 -0.0906 -0.482 0.025 -5.8971 0.74389 1.9911 0.0001 0.271 -0.0026 -0.0482 -0.492 ------------------------------------------------------------------------------- *** The minimum value given in the above-mentioned reference was F(X)=-4.682... This assure the necessity of using the simulated entropy (SE) techniques to secure our search for the global minimum. A. Sultan