【optimtool.unconstrain】无约束优化工具箱

# import packages
%matplotlib inline
import sympy as sp
import matplotlib.pyplot as plt
import optimtool as oo

def train(funcs, args, x_0):
    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)
    return colorlist, f_list, title

# 可视化函数：传参接口（颜色列表，函数值列表，标题列表）
def test(colorlist, f_list, title):
    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")
    return None

Extended Freudenstein & Roth function

$$f(x)=\sum _{i=1}^{n/2}(-13+x_{2i-1}+((5-x_{2i})x_{2i}-2)x_{2i}{)}^2+(-29+x_{2i-1}+((x_{2i}+1)x_{2i}-14)x_{2i}{)}^2,x_0=[0.5,-2,0.5,-2,...,0.5,-2].$$

# make data(4 dimension)
x = sp.symbols("x1:5")
f = (-13 + x[0] + ((5 - x[1])*x[1] - 2)*x[1])**2 + \
    (-29 + x[0] + ((x[1] + 1)*x[1] - 14)*x[1])**2 + \
    (-13 + x[2] + ((5 - x[3])*x[3] - 2)*x[3])**2 + \
    (-29 + x[2] + ((x[3] + 1)*x[3] - 14)*x[3])**2
x_0 = (1, -1, 1, -1) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Extended Trigonometric function:

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

# make data(2 dimension)
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 = (0.1, 0.1) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Extended Rosenbrock function

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

# 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 = (-2, 2, -2, 2) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Generalized Rosenbrock function

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

# make data(2 dimension)
x = sp.symbols("x1:3")
f = 100 * (x[1] - x[0]**2)**2 + (1 - x[0])**2
x_0 = (-1, 0.5) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Extended White & Holst function

$$f(x)=\sum _{i=1}^{n/2}c(x_{2i}-x_{2i-1}^3{)}^2+(1-x_{2i-1}{)}^2,x_0=[-1.2,1,...,-1.2,1].c=100$$

# make data(4 dimension)
x = sp.symbols("x1:5")
f = 100 * (x[1] - x[0]**3)**2 + \
    (1 - x[0])**2 + \
    100 * (x[3] - x[2]**3)**2 + \
    (1 - x[2])**2
x_0 = (-1, 0.5, -1, 0.5) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Extended Penalty function

$$f(x)=\sum _{i=1}^{n-1}(x_i-1{)}^2+(\sum _{j=1}^n{x}_j^2-0.25{)}^2,x_0=[1,2,...,n].$$

# make data(4 dimension)
x = sp.symbols("x1:5")
f = (x[0] - 1)**2 + (x[1] - 1)**2 + (x[2] - 1)**2 + \
    ((x[0]**2 + x[1]**2 + x[2]**2 + x[3]**2) - 0.25)**2
x_0 = (5, 5, 5, 5) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Perturbed Quadratic function

$$f(x)=\sum _{i=1}^{n}i{x}_i^2+\frac{1}{100}(\sum _{i=1}^n{x}_i{)}^2,x_0=[0.5,0.5,...,0.5].$$

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

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Raydan 1 function

$$f(x)=\sum _{i=1}^n\frac{i}{10}(\exp x_i-x_i),x_0=[1,1,...,1].$$

# make data(4 dimension)
x = sp.symbols("x1:5")
f = 0.1 * (sp.exp(x[0]) - x[0]) + \
    0.2 * (sp.exp(x[1]) - x[1]) + \
    0.3 * (sp.exp(x[2]) - x[2]) + \
    0.4 * (sp.exp(x[3]) - x[3])
x_0 = (0.5, 0.5, 0.5, 0.5) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Raydan 2 function

$$f(x)=\sum _{i=1}^n(\exp x_i-x_i),x_0=[1,1,...,1].$$

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

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Diagonal 1 function

$$f(x)=\sum _{i=1}^n(\exp x_i-i{x}_i),x_0=[1/n,1/n,...,1/n].$$

# make data(4 dimension)
x = sp.symbols("x1:5")
f = (sp.exp(x[0]) - x[0]) + \
    (sp.exp(x[1]) - 2 * x[1]) + \
    (sp.exp(x[2]) - 3 * x[2]) + \
    (sp.exp(x[3]) - 4 * x[3])
x_0 = (0.5, 0.5, 0.5, 0.5) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Diagonal 2 function

$$f(x)=\sum _{i=1}^n(\exp x_i-\frac{x_i}{i}),x_0=[1/1,1/2,...,1/n].$$

# make data(4 dimension)
x = sp.symbols("x1:5")
f = (sp.exp(x[0]) - x[0]) + \
    (sp.exp(x[1]) - x[1] / 2) + \
    (sp.exp(x[2]) - x[2] / 3) + \
    (sp.exp(x[3]) - x[3] / 4)
x_0 = (0.9, 0.6, 0.4, 0.3) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Diagonal 3 function

$$f(x)=\sum _{i=1}^n(\exp x_i-i\sin (x_i)),x_0=[1,1,...,1].$$

# make data(4 dimension)
x = sp.symbols("x1:5")
f = (sp.exp(x[0]) - sp.sin(x[0])) + \
    (sp.exp(x[1]) - 2 * sp.sin(x[1])) + \
    (sp.exp(x[2]) - 3 * sp.sin(x[2])) + \
    (sp.exp(x[3]) - 4 * sp.sin(x[3]))
x_0 = (0.5, 0.5, 0.5, 0.5) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Hager function

$$f(x)=\sum _{i=1}^n(\exp x_i-\sqrt{i}{x}_i),x_0=[1,1,...,1].$$

# make data(4 dimension)
x = sp.symbols("x1:5")
f = (sp.exp(x[0]) - x[0]) + \
    (sp.exp(x[1]) - sp.sqrt(2) * x[1]) + \
    (sp.exp(x[2]) - sp.sqrt(3) * x[2]) + \
    (sp.exp(x[3]) - sp.sqrt(4) * x[3])
x_0 = (0.5, 0.5, 0.5, 0.5) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Generalized Tridiagonal 1 function

$$f(x)=\sum _{i=1}^{n-1}(x_i+x_{i+1}-3{)}^2+(x_i-x_{i+1}+1{)}^4,x_0=[2,2,...,2].$$

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

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Extended Tridiagonal 1 function:

$$f(x)=\sum _{i=1}^{n/2}(x_{2i-1}+x_{2i}-3{)}^2+(x_{2i-1}-x_{2i}+1{)}^4,x_0=[2,2,...,2].$$

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

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)

Extended TET function : (Three exponential terms)

$$f(x)=\sum _{i=1}^{n/2}((\exp x_{2i-1}+3x_{2i}-0.1)+\exp (x_{2i-1}-3x_{2i}-0.1)+\exp (-x_{2i-1}-0.1)),x_0=[0.1,0.1,...,0.1].$$

# make data(4 dimension)
x = sp.symbols("x1:5")
f = sp.exp(x[0] + 3*x[1] - 0.1) + sp.exp(x[0] - 3*x[1] - 0.1) + sp.exp(-x[0] - 0.1) + \
    sp.exp(x[2] + 3*x[3] - 0.1) + sp.exp(x[2] - 3*x[3] - 0.1) + sp.exp(-x[2] - 0.1)
x_0 = (0.2, 0.2, 0.2, 0.2) # Random given

# train
color, values, title = train(funcs=f, args=x, x_0=x_0)

# test
test(color, values, title)