🚀 go-pugleaf

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