# README.md

`pip install optimtool`

## 1. 无约束优化算法性能对比

`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
import optimtool as oo

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 = oo.unconstrain.gradient_descent.barzilar_borwein(funcs, args, x_0, False, True)
f_list.append(f)
_, _, f = oo.unconstrain.newton.CG(funcs, args, x_0, False, True)
f_list.append(f)
_, _, f = oo.unconstrain.newton_quasi.L_BFGS(funcs, args, x_0, False, True)
f_list.append(f)
_, _, f = oo.unconstrain.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()```

## 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
import optimtool as oo

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 = oo.unconstrain.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 = oo.unconstrain.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()```

## 3. 等式约束优化测试

• from optimtool.constrain import equal

```import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

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 = oo.constrain.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 = oo.constrain.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()```

## 4. 不等式约束优化测试

• from optimtool.constrain import unequal

```import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

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 = oo.constrain.unequal.penalty_quadratic(funcs, args, cons, x_0, False, True, method="newton", sigma=10, epsilon=1e-6) # 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 = oo.constrain.unequal.penalty_interior_fraction(funcs, args, cons, x_0, False, True, method="newton")
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()```

`单独测试拉格朗日方法`

```# 导入符号运算的包
import sympy as sp

# 导入约束优化
import optimtool as oo

# 构造函数
f1 = sp.symbols("f1")
x1, x2, x3, x4 = sp.symbols("x1 x2 x3 x4")
f1 = x1**2 + x2**2 + 2*x3**3 + x4**2 - 5*x1 - 5*x2 - 21*x3 + 7*x4
c1 = 8 - x1 + x2 - x3 + x4 - x1**2 - x2**2 - x3**2 - x4**2
c2 = 10 + x1 + x4 - x1**2 - 2*x2**2 - x3**2 - 2*x4**2
c3 = 5 - 2*x1 + x2 + x4 - 2*x1**2 - x2**2 - x3**2
cons_unequal1 = sp.Matrix([c1, c2, c3])
funcs1 = sp.Matrix([f1])
args1 = sp.Matrix([x1, x2, x3, x4])
x_1 = (0, 0, 0, 0)

x_0, _, f = oo.constrain.unequal.lagrange_augmented(funcs1, args1, cons_unequal1, x_1, output_f=True, method="trust_region", sigma=1, muk=1, p=1.2)
for i in range(len(x_0)):
x_0[i] = round(x_0[i], 2)
print("\n最终收敛点：", x_0, "\n目标函数值：", f[-1])```

`result`

```最终收敛点： [ 2.5   2.5   1.87 -3.5 ]

## 5. 混合等式约束测试

• from optimtool.constrain import mixequal

L1罚函数法 `penalty_L1(funcs, args, cons_equal, cons_unequal, x_0, draw=True, output_f=False, method="gradient_descent", sigma=1, p=0.6, epsilon=1e-10, k=0)` mixequal.penalty_L1(funcs, args, cons_equal, cons_unequal, x_0)

```import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

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 = oo.constrain.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 = oo.constrain.mixequal.penalty_L1(funcs, args, cons_equal, cons_unequal, x_0, False, True)
f_list.append(f)
_, _, f = oo.constrain.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()```

## 6. 问题测试

• from optimtool.example import Lasso

```import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

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 = oo.example.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 = oo.example.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()```

## 7. WanYuan问题测试

• from optimtool.example import WanYuan

`问题描述`

`code`

```# 导包
import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

# 构造数据
m = 1
n = 2
a = 0.2
b = -1.4
c = 2.2
x3 = 2*(1/2)
y3 = 0
x_0 = (0, -1, -2.5, -0.5, 2.5, -0.05)

# 训练
oo.example.WanYuan.gauss_newton(1, 2, 0.2, -1.4, 2.2, 2**(1/2), 0, (0, -1, -2.5, -0.5, 2.5, -0.05), draw=True)```

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