基于复合优化加速算法研究实际问题

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

pip install optimtool>=2.5.0

加载hybird.nesterov.accer方法

import optimtool.hybrid as oh
nes_acc = oh.nesterov.accer

初始化输入数据

$$f(x)=\sum _{i=1}^n((n-\sum _{j=1}^n\cos x_j)+i(1-\cos x_i)-\sin x_i{)}^2,x_0=[0.2,0.2,...,0.2]$$

x = sp.symbols("x1:3")
f = (2 - (sp.cos(x[0]) + sp.cos(x[1])) + (1 - sp.cos(x[0])) - sp.sin(x[0]))**2 + \
    (2 - (sp.cos(x[0]) + sp.cos(x[1])) + 2 * (1 - sp.cos(x[1])) - sp.sin(x[1]))**2
x_0 = (0.2, 0.2) # Random given

查看复合优化算法的超参数

accer(funcs: FuncArray, args: ArgArray, x_0: PointArray, mu: float=1e-3, proxim: str=“L1”, lk: float=0.01, tk: float=0.02, verbose: bool=False, draw: bool=True, output_f: bool=False, epsilon: float=1e-6, k: int=0) -> OutputType

${\lambda }_k$为算法中的lk，$t_k$为算法中的tk，$\mathrm{prox}$算符的含义见如何研究带有不可微项的目标函数的局部极小值？。

调用并绘制迭代图

print(nes_acc(f, x, x_0, verbose=True, lk=0.1, tk=0.2))
# ensure lk and tk < 1 / L (L comes from lipschitz continuity condition)

默认的lk和tk能够解决稀疏矩阵解的问题，在实际问题中可以适当增大。

(0.2, 0.2)	0.033830304000793295	0
[0.19256431 0.19256431]	0.03265408936657944	1
[0.18638465 0.18638465]	0.031617693847704915	2
[0.17987066 0.17987066]	0.03047115719116562	3
[0.1730253 0.1730253]	0.029210872201568115	4
[0.16579189 0.16579189]	0.027822939438030186	5
[0.1581071 0.1581071]	0.02629215385584641	6
[0.14993051 0.14993051]	0.024608804452366514	7
[0.1412556 0.1412556]	0.02277288945465011	8
[0.13211227 0.13211227]	0.020796735113812578	9
[0.12256644 0.12256644]	0.018706522868828408	10
[0.11271834 0.11271834]	0.01654250102784132	11
[0.10269905 0.10269905]	0.014357303201991972	12
[0.09266423 0.09266423]	0.012211920332773922	13
[0.08278452 0.08278452]	0.01016943425860308	14
[0.07323288 0.07323288]	0.008287438822270246	15
[0.0641702 0.0641702]	0.006610792391354569	16
[0.05573159 0.05573159]	0.005166538950867488	17
[0.04801584 0.04801584]	0.003962275998264823	18
[0.04108007 0.04108007]	0.0029881260918894485	19
[0.03494014 0.03494014]	0.0022213296681673307	20
[0.0295762 0.0295762]	0.0016318641094921836	21
[0.02494136 0.02494136]	0.0011875956332260999	22
[0.02097132 0.02097132]	0.0008580603239475022	23
[0.01759323 0.01759323]	0.000616649225758882	24
[0.01473264 0.01473264]	0.0004414466868771702	25
[0.0123183 0.0123183]	0.0003151622104264512	26
[0.01028495 0.01028495]	0.0002245773826627801	27
[0.00857464 0.00857464]	0.00015981543713860277	28
[0.00713697 0.00713697]	0.00011361642729667025	29
[0.00592872 0.00592872]	8.070513699807076e-05	30
[0.00491322 0.00491322]	5.727893092786287e-05	31
[0.00405951 0.00405951]	4.061141417882402e-05	32
[0.00334156 0.00334156]	2.875475343419944e-05	33
[0.00273758 0.00273758]	2.032048717223606e-05	34
[0.00222925 0.00222925]	1.4320242141885246e-05	35
[0.00180129 0.00180129]	1.0050987097603078e-05	36
[0.00144084 0.00144084]	7.012839175347395e-06	37
[0.00113718 0.00113718]	4.85040116210444e-06	38
[0.00088126 0.00088126]	3.3109866323555416e-06	39
[0.00066554 0.00066554]	2.214909990704722e-06	40
[0.00048366 0.00048366]	1.434371414320055e-06	41
[0.00033027 0.00033027]	8.784544440346822e-07	42
[0.0002009 0.0002009]	4.82466914247575e-07	43
[9.1765239e-05 9.1765239e-05]	2.003667873131081e-07	44
[-3.10652223e-07 -3.10652223e-07]	6.2149745663463e-10	45

当$f$为非凸函数时，算法也收敛。