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优化算法研究Rastrigin函数

news 2025/10/7 7:10:40

使用optimtool库的方法

定义Rastrigin函数，本文重点研究方法的局部开发能力，函数在全局极小值附近有很多局部极小值，设置初始点为（0.5，-0.3），

import optimtool.unconstrain as ou
from optimtool.base import sp
x = sp.symbols("x1:3")
A = 10
sum_term = (x[0]**2 - A * sp.cos(2 * sp.pi * x[0])) + (x[1]**2 - A * sp.cos(2 * sp.pi * x[1]))
rastrigin = A * 2 + sum_term
x_0 = (0.5, -0.3)

信赖域方法的训练结果和可视化

ou.trust_region.steihaug_CG(rastrigin, x, x_0, verbose=True, draw=True)

(0.5, -0.3)     33.430169943749476      0
(0.5, -0.3)     33.430169943749476      1
(0.5, -0.3)     33.430169943749476      2
[ 0.49585852 -0.05003431]       20.735094807492853      3
[0.13531621 0.05308815] 3.9741075141269517      4
[-0.04558432 -0.00204971]       0.4102820555806389      5
[1.28201395e-03 1.12757902e-07] 0.00032606750446062375  6
[-2.75888328e-08 -3.63981485e-12]       1.5024374593316442e-13  7

在这里插入图片描述

L_BFGS训练结果和可视化

ou.newton_quasi.L_BFGS(rastrigin, x, x_0, verbose=True, draw=True)

(0.5, -0.3)     33.430169943749476      0
[ 0.49609375 -0.06423186]       21.05061587716868       1
[ 0.49540553 -0.03394355]       20.46897979380196       2
[0.49089607 0.00173255] 20.225218762652215      3
[0.490644   0.00120956] 20.223748098509294      4
[ 0.46559824 -0.02448895]       20.10282411064895       5
[ 0.15439562 -0.32475773]       19.003441057801915      6
[ 0.25376874 -0.21970435]       18.45738637665312       7
[ 0.3439645  -0.07807552]       16.87063276869083       8
[ 0.11471468 -0.17770738]       8.144098648171822       9
[ 0.0060908  -0.16881321]       5.153110356206469       10
[-0.00588779  0.00291503]       0.008562465730363785    11
[ 0.00050293 -0.00056868]       0.00011434117888674131  12
[2.32706011e-05 4.28990140e-05] 4.7253945653447976e-07  13
[-5.61985987e-06 -4.40772858e-06]       1.0120160730610205e-08  14
[ 4.14953189e-08 -2.53413392e-08]       4.666437717145493e-13   15
[-4.62768595e-10  5.97824247e-10]       1.1281917205678543e-16  16
[2.18391301e-13 3.53260480e-13] 3.4047712591437506e-23  17

在这里插入图片描述
可以看到信赖域方法和LBFGS方法能很好地跳过局部最优值20，现在设置初始点为（0.4,-0.3)，观察结果，

barzilar_borwein训练结果和可视化

x_0 = (0.4, -0.3)
ou.gradient_descent.barzilar_borwein(rastrigin, x, x_0, verbose=True, draw=True)

(0.4, -0.3)     31.43033988749895       0
[0.17185122 0.06495354] 6.13976171835116        1
[-0.20974784 -0.10661683]       9.714240474120007       2
[-0.01717213  0.01670725]       0.11377287169869504     3
[ 0.00486217 -0.00473276]       0.009133222904602967    4
[-6.53981222e-06  6.40427278e-06]       1.66220534617351e-08    5
[ 9.8504651e-10 -9.6463122e-10] 3.7520880612188806e-16  6
[-2.70225224e-19  2.64624797e-19]       2.823653449819599e-35   7

在这里插入图片描述

barzilar_borwein_ZhangHanger训练结果和可视化

x_0 = (0.4, -0.3)
ou.gradient_descent.barzilar_borwein(rastrigin, x, x_0, verbose=True, draw=True, method="ZhangHanger")

(0.4, -0.3)     31.43033988749895       0
[0.17185122 0.06495354] 6.13976171835116        1
[-0.05710822 -0.03798868]       0.9251057077803212      2
[0.00653816 0.00485932] 0.013163852514273213    3
[-1.02617867e-04 -7.68865311e-05]       3.261955745256119e-06   4
[2.37657913e-08 1.78088663e-08] 1.7409361445869945e-13  5
[-1.37967600e-15 -1.03385853e-15]       5.867221914198997e-28   6