[人工智能]【1.2.2】版本 optimtool使用文档

【1.2.2】版本 optimtool使用文档

版权归属：林景 （MIT Lisence）
pip install optimtool
相较于1.2.1版本的改变：

• 将Lasso问题求解的文件放在了example文件夹中
• 以后最优化的应用解法应该都放在example文件夹下
• unconstrain，constrain与hybrid用于更新底层算法，有好的可以加入，有不好的就删除。

1. 无约束优化算法性能对比

前五个参数完全一致，其中第四个参数是绘图接口，默认绘制单个算法的迭代过程；第五个参数是输出函数迭代值接口，默认为不输出。

method：用于传递线搜索方式

• from optimtool.unconstrain import gradient_descent

• from optimtool.unconstrain import newton

• from optimtool.unconstrain import newton_quasi

• from optimtool.unconstrain import trust_region

import sympy as sp
import matplotlib.pyplot as plt
from optimtool.unconstrain import gradient_descent, newton, newton_quasi, trust_region

f, x1, x2, x3, x4 = sp.symbols("f x1 x2 x3 x4")
f = (x1 - 1)**2 + (x2 - 1)**2 + (x3 - 1)**2 + (x1**2 + x2**2 + x3**2 + x4**2 - 0.25)**2
funcs = sp.Matrix([f])
args = sp.Matrix([x1, x2, x3, x4])
x_0 = (1, 2, 3, 4)

# 无约束优化测试函数性能对比
f_list = []
title = ["gradient_descent_barzilar_borwein", "newton_CG", "newton_quasi_L_BFGS", "trust_region_steihaug_CG"]
colorlist = ["maroon", "teal", "slateblue", "orange"]
_, _, f = gradient_descent.barzilar_borwein(funcs, args, x_0, False, True)
f_list.append(f)
_, _, f = newton.CG(funcs, args, x_0, False, True)
f_list.append(f)
_, _, f = newton_quasi.L_BFGS(funcs, args, x_0, False, True)
f_list.append(f)
_, _, f = trust_region.steihaug_CG(funcs, args, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

图像：

2. 非线性最小二乘问题

• from optimtool.unconstrain import nonlinear_least_square

method：用于传递线搜索方法

import sympy as sp
import matplotlib.pyplot as plt
from optimtool.unconstrain import nonlinear_least_square

r1, r2, x1, x2 = sp.symbols("r1 r2 x1 x2")
r1 = x1**3 - 2*x2**2 - 1
r2 = 2*x1 + x2 - 2
funcr = sp.Matrix([r1, r2])
args = sp.Matrix([x1, x2])
x_0 = (2, 2)

f_list = []
title = ["gauss_newton", "levenberg_marquardt"]
colorlist = ["maroon", "teal"]
_, _, f = nonlinear_least_square.gauss_newton(funcr, args, x_0, False, True) # 第五参数控制输出函数迭代值列表
f_list.append(f)
_, _, f = nonlinear_least_square.levenberg_marquardt(funcr, args, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

图示：

3. 等式约束优化测试

• from optimtool.constrain import equal

无约束内核默认采用wolfe线搜索方法

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
from optimtool.constrain import equal

f, x1, x2 = sp.symbols("f x1 x2")
f = x1 + np.sqrt(3) * x2
c1 = x1**2 + x2**2 - 1
funcs = sp.Matrix([f])
cons = sp.Matrix([c1])
args = sp.Matrix([x1, x2])
x_0 = (-1, -1)

f_list = []
title = ["penalty_quadratic", "lagrange_augmented"]
colorlist = ["maroon", "teal"]
_, _, f = equal.penalty_quadratic(funcs, args, cons, x_0, False, True) # 第四个参数控制单个算法不显示迭代图，第五参数控制输出函数迭代值列表
f_list.append(f)
_, _, f = equal.lagrange_augmented(funcs, args, cons, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

图示：

4. 不等式约束优化测试

• from optimtool.constrain import unequal

无约束内核默认采用wolfe线搜索方法

import sympy as sp
import matplotlib.pyplot as plt
from optimtool.constrain import unequal

f, x1, x2 = sp.symbols("f x1 x2")
f = x1**2 + (x2 - 2)**2
c1 = 1 - x1
c2 = 2 - x2
funcs = sp.Matrix([f])
cons = sp.Matrix([c1, c2])
args = sp.Matrix([x1, x2])
x_0 = (2, 3)

f_list = []
title = ["penalty_quadratic", "penalty_interior_fraction"]
colorlist = ["maroon", "teal"]
_, _, f = unequal.penalty_quadratic(funcs, args, cons, x_0, False, True) # 第四个参数控制单个算法不显示迭代图，第五参数控制输出函数迭代值列表
f_list.append(f)
_, _, f = unequal.penalty_interior_fraction(funcs, args, cons, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

图示：

5. 混合等式约束测试

• from optimtool.constrain import mixequal

无约束内核默认采用wolfe线搜索方法

import sympy as sp
import matplotlib.pyplot as plt
from optimtool.constrain import mixequal

f, x1, x2 = sp.symbols("f x1 x2")
f = (x1 - 2)**2 + (x2 - 1)**2
c1 = x1 - 2*x2
c2 = 0.25*x1**2 - x2**2 - 1
funcs = sp.Matrix([f])
cons_equal = sp.Matrix([c1])
cons_unequal = sp.Matrix([c2])
args = sp.Matrix([x1, x2])
x_0 = (0.5, 1)

f_list = []
title = ["penalty_quadratic", "penalty_L1", "lagrange_augmented"]
colorlist = ["maroon", "teal", "orange"]
_, _, f = mixequal.penalty_quadratic(funcs, args, cons_equal, cons_unequal, x_0, False, True) # 第四个参数控制单个算法不显示迭代图，第五参数控制输出函数迭代值列表
f_list.append(f)
_, _, f = mixequal.penalty_L1(funcs, args, cons_equal, cons_unequal, x_0, False, True)
f_list.append(f)
_, _, f = mixequal.lagrange_augmented(funcs, args, cons_equal, cons_unequal, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

图示：

6. Lasso问题测试

• from optimtool.example import Lasso

$$\min \frac{1}{2}||Ax?b|{|}^2+\mu ||x|{|}_1$$
给定$A_{m\times n}$，$x_{n\times 1}$，$b_{m\times 1}$，正则化常数$\mu$。解决该无约束最优化问题，该问题目标函数一阶不可导。

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
from optimtool.example import Lasso

import scipy.sparse as ss
f, A, b, mu = sp.symbols("f A b mu")
x = sp.symbols('x1:9')
m = 4
n = 8
u = (ss.rand(n, 1, 0.1)).toarray()
A = np.random.randn(m, n)
b = A.dot(u)
mu = 1e-2
args = sp.Matrix(x)
x_0 = tuple([1 for i in range(8)])

f_list = []
title = ["gradient_descent", "subgradient"]
colorlist = ["maroon", "teal"]
_, _, f = Lasso.gradient_descent(A, b, mu, args, x_0, False, True, epsilon=1e-4)# 第四个参数控制单个算法不显示迭代图，第五参数控制输出函数迭代值列表
f_list.append(f)
_, _, f = Lasso.subgradient(A, b, mu, args, x_0, False, True)
f_list.append(f)

# 绘图
handle = []
for j, z in zip(colorlist, f_list):
    ln, = plt.plot([i for i in range(len(z))], z, c=j, marker='o', linestyle='dashed')
    handle.append(ln)
plt.xlabel("$Iteration \ times \ (k)$")
plt.ylabel("$Objective \ function \ value: \ f(x_k)$")
plt.legend(handle, title)
plt.title("Performance Comparison")
plt.show()

图示：