NSGA-II works like magic!!

[dinohsu1019]

Hi Julian,

I am curious about how NSG-II works because I see things beyond my understanding.

This is a 4-objective 16-solution-variable discrete (integers) problem, here's the output of my code after each iteration:

n_gen=1 n_eval=60 opt_F=array([-0.15139894, -1.10365474, -1.96775103,  0.12905796]) opt_X_list=[1, 0, 4, 1, 0, 1, 0, 0, 0, 2, 1, 5, 4, 5, 1, 4] opt_index=1 hypervol=0.0
n_gen=2 n_eval=90 opt_F=array([-0.15139894, -1.10365474, -1.96775103,  0.12905796]) opt_X_list=[1, 0, 4, 1, 0, 1, 0, 0, 0, 2, 1, 5, 4, 5, 1, 4] opt_index=1 hypervol=0.0
n_gen=3 n_eval=120 opt_F=array([-0.15139894, -1.10365474, -1.96775103,  0.12905796]) opt_X_list=[1, 0, 4, 1, 0, 1, 0, 0, 0, 2, 1, 5, 4, 5, 1, 4] opt_index=1 hypervol=0.0
n_gen=4 n_eval=150 opt_F=array([-0.25025423, -1.3935039 , -2.39057435,  0.17497768]) opt_X_list=[2, 1, 0, 1, 1, 1, 0, 0, 0, 4, 1, 4, 1, 2, 3, 2] opt_index=12 hypervol=0.0
n_gen=5 n_eval=180 opt_F=array([-0.25025423, -1.3935039 , -2.39057435,  0.17497768]) opt_X_list=[2, 1, 0, 1, 1, 1, 0, 0, 0, 4, 1, 4, 1, 2, 3, 2] opt_index=9 hypervol=0.0
n_gen=6 n_eval=210 opt_F=array([-0.25025423, -1.3935039 , -2.39057435,  0.17497768]) opt_X_list=[2, 1, 0, 1, 1, 1, 0, 0, 0, 4, 1, 4, 1, 2, 3, 2] opt_index=7 hypervol=0.0
n_gen=7 n_eval=240 opt_F=array([-0.25025423, -1.3935039 , -2.39057435,  0.17497768]) opt_X_list=[2, 1, 0, 1, 1, 1, 0, 0, 0, 4, 1, 4, 1, 2, 3, 2] opt_index=7 hypervol=0.0
n_gen=8 n_eval=270 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=20 hypervol=0.0
n_gen=9 n_eval=300 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=20 hypervol=0.0
n_gen=10 n_eval=330 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=17 hypervol=0.0
n_gen=11 n_eval=360 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=14 hypervol=0.0
n_gen=12 n_eval=390 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=7 hypervol=0.0
n_gen=13 n_eval=420 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=7 hypervol=0.0
n_gen=14 n_eval=450 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=5 hypervol=0.0
n_gen=15 n_eval=480 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=3 hypervol=0.0
n_gen=16 n_eval=510 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=2 hypervol=0.0
n_gen=17 n_eval=540 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=25 hypervol=0.0
n_gen=18 n_eval=570 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=19 n_eval=600 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=20 n_eval=630 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=21 n_eval=660 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=22 n_eval=690 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=20 hypervol=0.0
n_gen=23 n_eval=720 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=24 n_eval=750 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=25 n_eval=780 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=26 n_eval=810 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=27 n_eval=840 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=28 n_eval=870 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=29 n_eval=900 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=0 hypervol=0.0
n_gen=30 n_eval=930 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=52 hypervol=0.0
n_gen=31 n_eval=960 opt_F=array([-0.28615907, -1.96680742, -3.25045177,  0.19348045]) opt_X_list=[6, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 1, 5, 0, 4] opt_index=46 hypervol=0.0
n_gen=32 n_eval=990 opt_F=array([-0.36443393, -1.63692172, -2.90660088,  0.18201871]) opt_X_list=[1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 4, 2, 2, 0, 4] opt_index=39 hypervol=0.0
n_gen=33 n_eval=1020 opt_F=array([-0.36443393, -1.63692172, -2.90660088,  0.18201871]) opt_X_list=[1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 4, 2, 2, 0, 4] opt_index=13 hypervol=0.0
n_gen=34 n_eval=1050 opt_F=array([-0.36618637, -1.7242604 , -3.15867877,  0.19520101]) opt_X_list=[4, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 4, 1, 2, 0, 4] opt_index=47 hypervol=0.0
n_gen=35 n_eval=1080 opt_F=array([-0.36618637, -1.7242604 , -3.15867877,  0.19520101]) opt_X_list=[4, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 4, 1, 2, 0, 4] opt_index=47 hypervol=0.0
n_gen=36 n_eval=1110 opt_F=array([-0.39393583, -1.82364553, -3.3285767 ,  0.19833308]) opt_X_list=[4, 1, 1, 1, 1, 0, 1, 0, 0, 3, 1, 4, 1, 5, 0, 1] opt_index=10 hypervol=0.0
n_gen=37 n_eval=1140 opt_F=array([-0.39393583, -1.82364553, -3.3285767 ,  0.19833308]) opt_X_list=[4, 1, 1, 1, 1, 0, 1, 0, 0, 3, 1, 4, 1, 5, 0, 1] opt_index=9 hypervol=0.0
n_gen=38 n_eval=1170 opt_F=array([-0.39393583, -1.82364553, -3.3285767 ,  0.19833308]) opt_X_list=[4, 1, 1, 1, 1, 0, 1, 0, 0, 3, 1, 4, 1, 5, 0, 1] opt_index=12 hypervol=0.0
n_gen=39 n_eval=1200 opt_F=array([-0.39393583, -1.82364553, -3.3285767 ,  0.19833308]) opt_X_list=[4, 1, 1, 1, 1, 0, 1, 0, 0, 3, 1, 4, 1, 5, 0, 1] opt_index=11 hypervol=0.0
n_gen=40 n_eval=1230 opt_F=array([-0.39393583, -1.82364553, -3.3285767 ,  0.19833308]) opt_X_list=[4, 1, 1, 1, 1, 0, 1, 0, 0, 3, 1, 4, 1, 5, 0, 1] opt_index=45 hypervol=0.0
n_gen=41 n_eval=1260 opt_F=array([-0.36618637, -1.7242604 , -3.15867877,  0.19520101]) opt_X_list=[4, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 4, 1, 2, 0, 4] opt_index=53 hypervol=0.0
n_gen=42 n_eval=1290 opt_F=array([-0.36618637, -1.7242604 , -3.15867877,  0.19520101]) opt_X_list=[4, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 4, 1, 2, 0, 4] opt_index=45 hypervol=0.0
n_gen=43 n_eval=1320 opt_F=array([-0.36618637, -1.7242604 , -3.15867877,  0.19520101]) opt_X_list=[4, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 4, 1, 2, 0, 4] opt_index=43 hypervol=0.0
n_gen=44 n_eval=1350 opt_F=array([-0.36618637, -1.7242604 , -3.15867877,  0.19520101]) opt_X_list=[4, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 4, 1, 2, 0, 4] opt_index=45 hypervol=0.0
n_gen=45 n_eval=1380 opt_F=array([-0.36618637, -1.7242604 , -3.15867877,  0.19520101]) opt_X_list=[4, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 4, 1, 2, 0, 4] opt_index=46 hypervol=0.0
n_gen=46 n_eval=1410 opt_F=array([-0.47693079, -1.89521156, -3.46582662,  0.19875202]) opt_X_list=[1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 4, 1, 5, 0, 4] opt_index=57 hypervol=0.0
n_gen=47 n_eval=1440 opt_F=array([-0.47693079, -1.89521156, -3.46582662,  0.19875202]) opt_X_list=[1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 4, 1, 5, 0, 4] opt_index=43 hypervol=0.0
n_gen=48 n_eval=1470 opt_F=array([-0.47693079, -1.89521156, -3.46582662,  0.19875202]) opt_X_list=[1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 4, 1, 5, 0, 4] opt_index=45 hypervol=0.0
n_gen=49 n_eval=1500 opt_F=array([-0.47693079, -1.89521156, -3.46582662,  0.19875202]) opt_X_list=[1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 4, 1, 5, 0, 4] opt_index=36 hypervol=0.0
n_gen=50 n_eval=1530 opt_F=array([-0.47693079, -1.89521156, -3.46582662,  0.19875202]) opt_X_list=[1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 4, 1, 5, 0, 4] opt_index=36 hypervol=0.0

Here's algorithm's output:

==========================================================
n_gen  |  n_eval  | n_nds  |      eps      |   indicator  
==========================================================
     1 |       60 |      5 |             - |             -
     2 |       90 |      9 |  0.0866679452 |         ideal
     3 |      120 |     12 |  0.2155194480 |         ideal
     4 |      150 |     14 |  0.0393967788 |         ideal
     5 |      180 |     17 |  0.0585303336 |         ideal
     6 |      210 |     19 |  0.0676637076 |         ideal
     7 |      240 |     23 |  0.0188717395 |             f
     8 |      270 |     23 |  0.0186394568 |         ideal
     9 |      300 |     27 |  0.0323661496 |         ideal
    10 |      330 |     30 |  0.0144949961 |         ideal
    11 |      360 |     33 |  0.0142957614 |             f
    12 |      390 |     24 |  0.0141490101 |             f
    13 |      420 |     24 |  0.0149806898 |             f
    14 |      450 |     21 |  0.0105063242 |             f
    15 |      480 |     23 |  0.0782577708 |         nadir
    16 |      510 |     24 |  0.0040887717 |             f
    17 |      540 |     26 |  0.0018889471 |             f
    18 |      570 |     31 |  0.0842704954 |         ideal
    19 |      600 |     31 |  0.000000E+00 |             f
    20 |      630 |     33 |  0.0879301710 |         ideal
    21 |      660 |     39 |  0.0442056610 |         ideal
    22 |      690 |     42 |  0.0022914463 |             f
    23 |      720 |     48 |  0.0088060016 |             f
    24 |      750 |     41 |  0.0017797180 |             f
    25 |      780 |     43 |  0.0020618241 |             f
    26 |      810 |     47 |  0.0072848442 |             f
    27 |      840 |     49 |  0.0966954521 |         nadir
    28 |      870 |     58 |  0.0308663478 |         ideal
    29 |      900 |     58 |  0.0036456507 |             f
    30 |      930 |     60 |  0.0050430042 |             f
    31 |      960 |     60 |  0.0059040738 |         ideal
    32 |      990 |     60 |  0.0040887592 |             f
    33 |     1020 |     60 |  0.0017069884 |             f
    34 |     1050 |     60 |  0.0069860236 |         ideal
    35 |     1080 |     60 |  0.0133165835 |         nadir
    36 |     1110 |     60 |  0.0023566502 |             f
    37 |     1140 |     60 |  0.0944882747 |         nadir
    38 |     1170 |     60 |  0.0019453561 |             f
    39 |     1200 |     60 |  0.0046398226 |         ideal
    40 |     1230 |     60 |  0.0024482006 |             f
    41 |     1260 |     60 |  0.0076836134 |             f
    42 |     1290 |     60 |  0.0031854379 |             f
    43 |     1320 |     60 |  0.0553813194 |         ideal
    44 |     1350 |     60 |  0.000000E+00 |             f
    45 |     1380 |     60 |  0.0027243435 |         ideal
    46 |     1410 |     59 |  0.0031534453 |             f
    47 |     1440 |     60 |  0.0025622886 |             f
    48 |     1470 |     60 |  8.139130E-06 |             f
    49 |     1500 |     53 |  0.0059790858 |         ideal
    50 |     1530 |     54 |  0.1239611146 |         nadir

My code displays the optimal solution chosen from the NDS after that iteration, you can see f4 goes very close to 0.2 but not above it while increasing f1, f2, and f3. The thing is f4<0.2 is not a constraint at all, the algorithm doesn't know 0.2, the 0.2 is the criteria that I use to choose the optimal one from NDS.
To be clear about the criteria:
f1: -annual return
f2: -annual sharpe
f3: -annual sortino
f4: -mdd (max drawdown)
All solutions are candidate trading strategies as combinations of conditions. the absolute value of mdd (-f4) is a major concern since it represents the maximum possible loss of the strategy, so I want to control it under 0.2 (i.e., 20%) and maximize annual return and annual sortino, so I define the selection criteria as:
annual_return * annual_sortino / (abs(IDEAL_MDD/2)*10)**2, if mdd>IDEAL_MDD
annual_return * annual_sortino / (abs(mdd)*10)**2, otherwise
For now, IDEAL_MDD = -0.2. the (*10 and **2 is to double the impact of mdd when it is above the ideal level.
Note that annual_return is very sensitive to mdd, I remember if I reduce abs(mdd) from 0.25 to 0.2, annual return goes from 0.5 down to 0.3. But with this algorith, abs(mdd) can stay below 0.2 and still increase annual return and annual sortino to a satisfactory level.

How does NSGA-II algorithm achieve this not being aware of my selection criteria?

Thanks again.

[dinohsu1019]

As a further observation, the IDEAL_MDD=-0.2 (corresponding to objective 4) works so well is a coindincidence that in the nds, there are many solutions that go around -IDEAL_MDD=0.2 area to choose from, if IDEAL_MDD is, ininstead, -0.16 or -018, there are not many solutions to choose from and the performance of the optimal one is not as good.

[blankjul]

Someone else might know a little more about your application.

Unfortunatly, I am not able to provide detailed input on applications due to my own time limitation.

[dinohsu1019]

That's fine, thank you.