import optimtool as oo
from optimtool.base import np, sp, plt

pip install optimtool>=2.4.2

加载barzilar_borwein算法

import optimtool.unconstrain as ou
barzilar_borwein = ou.gradient_descent.barzilar_borwein

初始化输入数据

$$f(x)=\sum _{i=1}^{n/2}c(x_{2i}-x_{2i-1}^2{)}^2+(1-x_{2i-1}{)}^2,x_0=[-1.2,1,...,-1.2,1].c=100$$

# make data(4 dimension)
x = sp.symbols("x1:5")
f = 100 * (x[1] - x[0]**2)**2 + \
    (1 - x[0])**2 + \
    100 * (x[3] - x[2]**2)**2 + \
    (1 - x[2])**2
x_0 = (-1.2, 1, -1.2, 1) # Random given

采用默认配置研究

print(f, x, x_0, verbose=True)

出现如下报错：

Traceback (most recent call last):
  File "E:\workspace\test.py", line 10, in <module>
    print(barzilar_borwein(f, x, x_0, verbose=True))
  File "D:\anaconda\lib\site-packages\optimtool\unconstrain\gradient_descent.py", line 167, in barzilar_borwein
    alpha = search(funcs, args, x_0, dk, k, point, c1, beta, alpha, constant)
  File "D:\anaconda\lib\site-packages\optimtool\_search.py", line 190, in Grippo
    assert alpha > 0
AssertionError

意思是罚因子项小了些，需要把基于Grippo的算法的约束因子微调大些，因为只有alpha>0才会梯度下降。

微调算法默认参数

print(barzilar_borwein(f, x, x_0, verbose=True, c1=0.9, beta=0.9, M=10))

(-1.2, 1, -1.2, 1)	48.4	0
[-1.17218609  1.01135262 -1.17218609  1.01135262]	35.74234399993523	1
[-1.07573959  1.05146734 -1.07573959  1.05146734]	10.853930877079154	2
[-1.03729194  1.06784358 -1.03729194  1.06784358]	8.314339559517752	3
[-1.03059542  1.06930565 -1.03059542  1.06930565]	8.256942356362453	4
[-1.02949345  1.06786974 -1.02949345  1.06786974]	8.250528895172161	5
[-1.01145948  1.02980508 -1.01145948  1.02980508]	8.101063953719645	6
[ 0.18155041 -0.21954138  0.18155041 -0.21954138]	14.091164315347264	7
[0.06257751 0.14023334 0.06257751 0.14023334]	5.474008089570103	8
[0.08739254 0.0122698  0.08739254 0.0122698 ]	1.6699964827703286	9
[0.09717293 0.00770988 0.09717293 0.00770988]	1.6307938771999826	10
[0.12174316 0.01260807 0.12174316 0.01260807]	1.5436499168969129	11
[0.71993385 0.17321525 0.71993385 0.17321525]	23.974225961732103	12
[0.69341458 0.19173749 0.69341458 0.19173749]	16.902165798059535	13
[0.59947343 0.25999758 0.59947343 0.25999758]	2.2957547438866586	14
[0.56017728 0.29391331 0.56017728 0.29391331]	0.4659728719309865	15
[0.55278525 0.30213422 0.55278525 0.30213422]	0.4023650853432392	16
[0.55308681 0.30367679 0.55308681 0.30367679]	0.4004558072779859	17
[0.55647063 0.30743861 0.55647063 0.30743861]	0.3944231285815026	18
[0.86021216 0.65100256 0.86021216 0.65100256]	1.6219430382036113	19
[0.70931846 0.73951826 0.70931846 0.73951826]	11.344620053720568	20
[0.8100935  0.66909219 0.8100935  0.66909219]	0.10510577406964994	21
[0.81731167 0.6650097  0.81731167 0.6650097 ]	0.06853647456044182	22
[0.8164723  0.66582992 0.8164723  0.66582992]	0.0674919060017248	23
[0.81662502 0.66605803 0.81662502 0.66605803]	0.06738671804221721	24
[0.86835509 0.75122239 0.86835509 0.75122239]	0.03624918137947759	25
[0.77235521 0.82683779 0.77235521 0.82683779]	10.711743471603352	26
[0.81192998 0.80138115 0.81192998 0.80138115]	4.1121138687376	27
[0.88346099 0.75768716 0.88346099 0.75768716]	0.1312782107449447	28
[0.87314179 0.76370122 0.87314179 0.76370122]	0.0325369376654432	29
[0.87401353 0.76337883 0.87401353 0.76337883]	0.03179943209268424	30
[0.87410199 0.76351067 0.87410199 0.76351067]	0.031759723095652316	31
[0.88694234 0.78612821 0.88694234 0.78612821]	0.025622065675204374	32
[0.95184857 0.98548447 0.95184857 0.98548447]	1.267694088111409	33
[1.03892769 0.93988749 1.03892769 0.93988749]	3.8941464214885255	34
[0.98107815 0.96769113 0.98107815 0.96769113]	0.006075928176673569	35
[0.9830691  0.96669501 0.9830691  0.96669501]	0.0005879072914185411	36
[0.98321319 0.96663944 0.98321319 0.96663944]	0.0005645390367161203	37
[0.98321997 0.96665369 0.98321997 0.96665369]	0.0005640588458026667	38
[0.98971109 0.97943603 0.98971109 0.97943603]	0.0002134165931058616	39
[0.95171388 1.02355346 0.95171388 1.02355346]	2.779755313116296	40
[0.98444683 1.00639357 0.98444683 1.00639357]	0.2781156378581681	41
[1.00030797 0.99835475 1.00030797 0.99835475]	0.001022880219129762	42
[0.99938771 0.99881442 0.99938771 0.99881442]	1.0480525859292297e-06	43
[0.99940435 0.99880671 0.99940435 0.99880671]	7.106978129659511e-07	44
[0.9994046  0.99880718 0.9994046  0.99880718]	7.101292197589673e-07	45
[0.99953764 0.99907365 0.99953764 0.99907365]	4.282354083211168e-07	46
[1.00000356 0.9999974  1.00000356 0.9999974 ]	1.8955821269642996e-08	47
[0.99066961 1.00465583 0.99066961 1.00465583]	0.10809652952210036	48
[0.9999825  0.99996504 0.9999825  0.99996504]	6.131958176710077e-10	49
[0.99998255 0.99996503 0.99998255 0.99996503]	6.100493838834033e-10	50
[0.99998256 0.99996504 0.99998256 0.99996504]	6.095180589185439e-10	51
[0.9999865  0.99997287 0.9999865  0.99997287]	3.676877863890199e-10	52
[0.99993462 1.00003166 0.99993462 1.00003166]	5.284526365615982e-06	53
[1.00001028 0.99999391 1.00001028 0.99999391]	1.4234808255557074e-07	54
[0.99999962 0.99999923 0.99999962 0.99999923]	2.964397028197504e-13	55
[0.99999962 0.99999923 0.99999962 0.99999923]	2.9534013511438936e-13	56
[0.99999962 0.99999923 0.99999962 0.99999923]	2.9507256358442176e-13	57
[1. 1. 1. 1.]	1.88063786036214e-20	58
[1.00000001 1.         1.00000001 1.        ]	1.0948317471092592e-13	59
[1. 1. 1. 1.]	1.1024327807665554e-22	60
[1. 1. 1. 1.]	3.549874073494553e-30	61

另一种步长搜索器

barzilar_borwein(f, x, x_0, verbose=True, method="ZhangHanger", c1=0.8, beta=0.8, eta=0.6)

(-1.2, 1, -1.2, 1)	48.4	0
[-1.14402704  1.02284611 -1.14402704  1.02284611]	25.547385906612575	1
[-1.04917293  1.06298694 -1.04917293  1.06298694]	8.683638321782801	2
[-1.03278859  1.06919128 -1.03278859  1.06919128]	8.265748234572825	3
[-1.02996132  1.06871536 -1.02996132  1.06871536]	8.253952272254296	4
[-1.02880459  1.06645286 -1.02880459  1.06645286]	8.244940882294298	5
[-0.75881417  0.49681826 -0.75881417  0.49681826]	7.434444489232861	6
[-0.71206416  0.52368118 -0.71206416  0.52368118]	5.917743983089465	7
[-0.71412212  0.51847911 -0.71412212  0.51847911]	5.890908900222328	8
[-0.71176181  0.51445338 -0.71176181  0.51445338]	5.872576785186908	9
[-0.38846072  0.08764221 -0.38846072  0.08764221]	4.655999792942503	10
[-0.34569326  0.13056384 -0.34569326  0.13056384]	3.646245482183912	11
[-0.34190925  0.12336077 -0.34190925  0.12336077]	3.609784188103343	12
[-0.31361399  0.10306018 -0.31361399  0.10306018]	3.4555935448716233	13
[ 0.00397922 -0.04371105  0.00397922 -0.04371105]	2.3665227959068664	14
[0.02130878 0.03512281 0.02130878 0.03512281]	2.1560573397774148	15
[ 0.03323879 -0.00159448  0.03323879 -0.00159448]	1.8707117086887108	16
[0.04338918 0.00129321 0.04338918 0.00129321]	1.8302779870130716	17
[0.22543897 0.01257035 0.22543897 0.01257035]	1.4925384627561797	18
[0.21426361 0.05756136 0.21426361 0.05756136]	1.2619193148303254	19
[0.22533239 0.04752472 0.22533239 0.04752472]	1.2023322716998874	20
[0.23164194 0.05078892 0.23164194 0.05078892]	1.1823945212759244	21
[0.39858439 0.12616496 0.39858439 0.12616496]	0.9373190719661812	22
[0.38145931 0.15408892 0.38145931 0.15408892]	0.7799006036978148	23
[0.38942728 0.14871975 0.38942728 0.14871975]	0.7473195943947577	24
[0.39208339 0.15075934 0.39208339 0.15075934]	0.740889445363389	25
[0.54384873 0.27095384 0.54384873 0.27095384]	0.5393305593466469	26
[0.52980061 0.28649559 0.52980061 0.28649559]	0.4489189523781992	27
[0.53484966 0.28379459 0.53484966 0.28379459]	0.4337598682697777	28
[0.53596463 0.28493254 0.53596463 0.28493254]	0.43173927738659845	29
[0.66711966 0.42695983 0.66711966 0.42695983]	0.28705969635936956	30
[0.6555018  0.43706044 0.6555018  0.43706044]	0.24824451482332732	31
[0.66027465 0.43437596 0.66027465 0.43437596]	0.23133012100250183	32
[0.6607599 0.4349673 0.6607599 0.4349673]	0.2307032124435715	33
[0.76387974 0.57216118 0.76387974 0.57216118]	0.13727493959232465	34
[0.75426998 0.57944271 0.75426998 0.57944271]	0.14289849619262165	35
[0.75982798 0.57625237 0.75982798 0.57625237]	0.1156011583706373	36
[0.76005697 0.57658352 0.76005697 0.57658352]	0.11538866972480637	37
[0.86143683 0.73133553 0.86143683 0.73133553]	0.061459919065018784	38
[0.85446275 0.73571119 0.85446275 0.73571119]	0.04864447302661772	39
[0.85725529 0.73429265 0.85725529 0.73429265]	0.040822666441657225	40
[0.85735963 0.73444416 0.85735963 0.73444416]	0.04076977055273804	41
[0.87135784 0.75854423 0.87135784 0.75854423]	0.033201365652168045	42
[0.87174017 0.76736276 0.87174017 0.76736276]	0.04394760040221391	43
[0.87398087 0.76619333 0.87398087 0.76619333]	0.03286687198523424	44
[0.8753094  0.76561167 0.8753094  0.76561167]	0.031157068795362352	45
[0.87537792 0.76574967 0.87537792 0.76574967]	0.031118963457695742	46
[0.89466678 0.79955226 0.89466678 0.79955226]	0.022343784290254107	47
[0.89301787 0.80235929 0.89301787 0.80235929]	0.02765007340992803	48
[0.89462001 0.80156035 0.89462001 0.80156035]	0.02250531824469297	49
[0.89539064 0.80127024 0.89539064 0.80127024]	0.021927489016178737	50
[0.89544715 0.80138049 0.89544715 0.80138049]	0.021902219932405945	51
[0.92813781 0.85996455 0.92813781 0.85996455]	0.010763616666667527	52
[0.92614688 0.86141869 0.92614688 0.86141869]	0.013603307269649175	53
[0.92787507 0.8605771  0.92787507 0.8605771 ]	0.01043214239830986	54
[0.92788077 0.86066166 0.92788077 0.86066166]	0.010420494397709812	55
[0.928094   0.86105668 0.928094   0.86105668]	0.010359161042322382	56
[0.9705483  0.94169756 0.9705483  0.94169756]	0.001749003669665157	57
[0.96879935 0.94379027 0.96879935 0.94379027]	0.007392649706546043	58
[0.96964345 0.94336767 0.96964345 0.94336767]	0.003839211350773539	59
[0.97099486 0.9427037  0.97099486 0.9427037 ]	0.0016858380908765503	60
[0.97100383 0.9427304  0.97100383 0.9427304 ]	0.0016843424445882975	61
[0.97155834 0.94380875 0.97155834 0.94380875]	0.0016205873167718177	62
[0.99298678 0.98746597 0.99298678 0.98746597]	0.0005149481798177243	63
[0.99405866 0.98693913 0.99405866 0.98693913]	0.0003651100160972216	64
[0.99358383 0.987184   0.99358383 0.987184  ]	8.245778054657986e-05	65
[0.99358682 0.987189   0.99358682 0.987189  ]	8.239052921973421e-05	66
[0.99405188 0.98811496 0.99405188 0.98811496]	7.087718695175268e-05	67
[0.99968373 1.0000437  0.99968373 1.0000437 ]	9.163261199483933e-05	68
[1.00005465 0.99985861 1.00005465 0.99985861]	1.2575323104527597e-05	69
[0.99995439 0.99990868 0.99995439 0.99990868]	4.162393555661443e-09	70
[0.99995444 0.9999087  0.99995444 0.9999087 ]	4.1577523758631505e-09	71
[0.99995448 0.99990879 0.99995448 0.99990879]	4.150285949048248e-09	72
[0.99999998 1.         0.99999998 1.        ]	1.6347843490821497e-13	73
[1.00002865 0.9999857  1.00002865 0.9999857 ]	1.027136418355962e-06	74
[1. 1. 1. 1.]	6.014856540114234e-20	75
[1. 1. 1. 1.]	5.359465147840669e-20	76
[1. 1. 1. 1.]	5.3551774107473235e-20	77
[1. 1. 1. 1.]	1.225496980276194e-21	78