Algorithm purpose

   The minimum value (minimum value) for solving unconstrained optimization problems.

Algorithm features

  There is no need to calculate the derivative of the objective function in the algorithm step.

Algorithm steps

Example 1

  Solve $\min\ f(x):=(1-x_1)^2+5{(x_2-{x_1}^2)}^2.$min f(x):=(1−x1​)2+5(x2​−x1​2)2.
Solution: Set the initial point to $(2,0)^T$(2,0)T, Initial detection step $\delta = \frac{1}{2}$δ=21​, Acceleration factor $\alpha=1$α=1, Attenuation factor $\beta=\frac{1}{2}$β=21​,Allowable error $\epsilon = 0.2$ϵ=0.2。
  The Python code solution is as follows:

import numpy as np


def function1(x):
    return (1 - x[0]) ** 2 + 5 * (x[1] - x[0] ** 2) ** 2


# Enter the initial detection search step delta, acceleration factor alpha(alpha>=1), reduction rate beta(0<beta<1), allowable error epsilon(epsilon>0), initial point xk
delta, alpha, beta, epsilon, xk = 0.5, 1, 0.5, 0.2, np.array([2, 0])
yk = xk.copy()

# Find the dimension of the problem
dim = len(xk)

# Initialize the number of iterations
k = 1

while delta > epsilon:
    # Output the basic information of this search
    print('Enter the first', k, 'Round iteration')
    print('Base point:', xk)
    print('Function value at base point:', function1(xk))
    print('The starting point for detection is:', yk)
    print('Detect the function value at the starting point', function1(yk))
    print('Probing search step delta:', delta)

    # Enter detection movement
    for i in range(dim):
        # Generate the coordinate direction of this detection
        e = np.zeros([1, dim])[0]
        e[i] = 1

        # Calculate the detected points
        t1, t2 = function1(yk + delta * e), function1(yk)
        if t1 < t2:
            yk = yk + delta * e
        else:
            t1, t2 = function1(yk - delta * e), function1(yk)
            if t1 < t2:
                yk = yk - delta * e
        print('Th', i + 1, 'The point obtained by this detection is', yk)
        print('Function value corresponding to this point', function1(yk))

    # Determine the new base point and calculate the new detection initial point
    t1, t2 = function1(yk), function1(xk)
    if t1 < t2:
        xk, yk = yk, yk + alpha * (yk - xk)
    else:
        delta, yk = delta * beta, xk
    k += 1

    print("\n")

  Result

Enter the 1  Round iteration
 Base point: [2 0]
 Function value at base point: 81
 The starting point for detection is: [2 0]
 Probe the function value at the starting point 81
 Probe search step delta: 0.5
 First 1  The points obtained by this detection are [1.5 0. ]
 Function value corresponding to this point 25.5625
 First 2  The points obtained by this detection are [1.5 0.5]
 Function value corresponding to this point 15.5625

 Enter the 2  Round iteration
 Base point: [1.5 0.5]
 Function value at base point: 15.5625
 The starting point for detection is: [1. 1.]
 Probe the function value at the starting point 0.0
 Probe search step delta: 0.5
 First 1  The points obtained by this detection are [1. 1.]
 Function value corresponding to this point 0.0
 First 2  The points obtained by this detection are [1. 1.]
 Function value corresponding to this point 0.0

 Enter the 3  Round iteration
 Base point: [1. 1.]
 Function value at base point: 0.0
 The starting point for detection is: [0.5 1.5]
 Probe the function value at the starting point 8.0625
 Probe search step delta: 0.5
 First 1  The points obtained by this detection are [1.  1.5]
 Function value corresponding to this point 1.25
 First 2  The points obtained by this detection are [1. 1.]
 Function value corresponding to this point 0.0

 Enter the 4  Round iteration
 Base point: [1. 1.]
 Function value at base point: 0.0
 The starting point for detection is: [1. 1.]
 Probe the function value at the starting point 0.0
 Probe search step delta: 0.25
 First 1  The points obtained by this detection are [1. 1.]
 Function value corresponding to this point 0.0
 First 2  The points obtained by this detection are [1. 1.]
 Function value corresponding to this point 0.0

So the approximate minimum point is $(1,1)^T$(1,1)T, The corresponding minimum value is $0$0。

Example 2

  Solve $\min f(x):={x_1}^2+{x_2}^2-4x_1+2x_2+7.$minf(x):=x1​2+x2​2−4x1​+2x2​+7.
Solution: Set the initial point to $(0,0)^T$(0,0)T, The initial detection step $\delta = 1$δ=1, Acceleration factor $\alpha = 1$α=1, Attenuation factor $\beta = 0.25$β=0.25,Allowable error $\epsilon = 0.05$ϵ=0.05。
  The Python code is as follows:

import numpy as np


def function1(x):
    return x[0] ** 2 + x[1] ** 2 - 4 * x[0] + 2 * x[1] + 7


# Enter the initial detection search step delta, acceleration factor alpha(alpha>=1), reduction rate beta(0<beta<1), allowable error epsilon(epsilon>0), initial point xk
delta, alpha, beta, epsilon, xk = 1, 1, 0.25, 0.05, np.array([0, 0])
yk = xk.copy()

# Find the dimension of the problem
dim = len(xk)

# Initialize the number of iterations
k = 1

while delta > epsilon:
    # Output the basic information of this search
    print('Enter the first', k, 'Round iteration')
    print('Base point:', xk)
    print('Function value at base point:', function1(xk))
    print('The starting point for detection is:', yk)
    print('Detect the function value at the starting point', function1(yk))
    print('Probing search step delta:', delta)

    # Enter detection movement
    for i in range(dim):
        # Generate the coordinate direction of this detection
        e = np.zeros([1, dim])[0]
        e[i] = 1

        # Calculate the detected points
        t1, t2 = function1(yk + delta * e), function1(yk)
        if t1 < t2:
            yk = yk + delta * e
        else:
            t1, t2 = function1(yk - delta * e), function1(yk)
            if t1 < t2:
                yk = yk - delta * e
        print('Th', i + 1, 'The point obtained by this detection is', yk)
        print('Function value corresponding to this point', function1(yk))

    # Determine the new base point and calculate the new detection initial point
    t1, t2 = function1(yk), function1(xk)
    if t1 < t2:
        xk, yk = yk, yk + alpha * (yk - xk)
    else:
        delta, yk = delta * beta, xk
    k += 1

    print("\n")

  Result:

Enter the 1  Round iteration
 Base point: [0 0]
 Function value at base point: 7
 The starting point for detection is: [0 0]
 Probe the function value at the starting point 7
 Probe search step delta: 1
 First 1  The points obtained by this detection are [1. 0.]
 Function value corresponding to this point 4.0
 First 2  The points obtained by this detection are [ 1. -1.]
 Function value corresponding to this point 3.0

 Enter the 2  Round iteration
 Base point: [ 1. -1.]
 Function value at base point: 3.0
 The starting point for detection is: [ 2. -2.]
 Probe the function value at the starting point 3.0
 Probe search step delta: 1
 First 1  The points obtained by this detection are [ 2. -2.]
 Function value corresponding to this point 3.0
 First 2  The points obtained by this detection are [ 2. -1.]
 Function value corresponding to this point 2.0

 Enter the 3  Round iteration
 Base point: [ 2. -1.]
 Function value at base point: 2.0
 The starting point for detection is: [ 3. -1.]
 Probe the function value at the starting point 3.0
 Probe search step delta: 1
 First 1  The points obtained by this detection are [ 2. -1.]
 Function value corresponding to this point 2.0
 First 2  The points obtained by this detection are [ 2. -1.]
 Function value corresponding to this point 2.0

 Enter the 4  Round iteration
 Base point: [ 2. -1.]
 Function value at base point: 2.0
 The starting point for detection is: [ 2. -1.]
 Probe the function value at the starting point 2.0
 Probe search step delta: 0.25
 First 1  The points obtained by this detection are [ 2. -1.]
 Function value corresponding to this point 2.0
 First 2  The points obtained by this detection are [ 2. -1.]
 Function value corresponding to this point 2.0

 Enter the 5  Round iteration
 Base point: [ 2. -1.]
 Function value at base point: 2.0
 The starting point for detection is: [ 2. -1.]
 Probe the function value at the starting point 2.0
 Probe search step delta: 0.0625
 First 1  The points obtained by this detection are [ 2. -1.]
 Function value corresponding to this point 2.0
 First 2  The points obtained by this detection are [ 2. -1.]
 Function value corresponding to this point 2.0

So the approximate minimum point obtained is $(2,-1)^T$(2,−1)T, The corresponding minimum value is $2$2。