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Author: sichkar-valentyn

Codes/Function_Optimization.py

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+# Implementing the task
+# Optimization of smooth and non-smooth functions
+
+
+import numpy as np
+import matplotlib.pyplot as plt
+from scipy import optimize
+
+
+# Task 1 - smooth function
+# Initial smooth function
+# f(x) = sin(x / 5) * exp(x / 10) + 5 * exp(-x / 2)
+# Function returns array with 'y' results from input array of 'x' for plotting by using 'np' instead of 'math'
+def f_array(k):
+    return np.sin(k / 5) * np.exp(k / 10) + 5 * np.exp(-k / 2)
+
+
+# Finding minimum of smooth function with 'BFGS' method
+# Setting initial point in form of 'ndarray' as 'minimize' function requires it in form of 'ndarray'
+x0_start = np.array([2])
+# print(type(x))  -->  <class 'numpy.ndarray'>
+# print(x.shape)  -->  (1,)
+# Finding minimum of the function from starting point 'x_start = 2'
+# Using 'minimize' function from 'scipy.optimize' library
+y0_min = optimize.minimize(f_array, x0_start, method='BFGS')
+print(y0_min)  # fun = 1.7452682903449388, iterations = 6
+print()
+
+
+# Setting initial point in form of 'ndarray' as 'minimize' function requires it in form of 'ndarray'
+x1_start = np.array([30])
+# print(type(x))  -->  <class 'numpy.ndarray'>
+# print(x.shape)  -->  (1,)
+# Finding minimum of smooth function from starting point 'x_start = 30'
+# Using 'minimize' function from 'scipy.optimize' library
+y1_min = optimize.minimize(f_array, x1_start, method='BFGS')
+print(y1_min)  # fun = -11.898894665981285, iterations = 6
+print()
+
+
+# Finding minimum of smooth function with 'differential evolution' method
+# Searching in the range [1, 30]
+# Setting the range for searching in form of tuple inside list as function requires it
+x2_range = [(1, 30)]  # tuple inside list
+# Finding minimum of smooth function
+y2_min = optimize.differential_evolution(f_array, [(1, 30)])
+print(y2_min)  # fun = -11.89889467, iterations = 5
+print()
+
+
+# Task 2 - non-smooth function
+# Initial non-smooth function
+# We take here the same function but only with integer results
+# By using 'np.int_' we return 'numpy.ndarray' of integer numbers
+def h_array(k):
+    return np.int_(f_array(k))
+
+
+# Finding minimum of non-smooth function with 'BFGS' method
+# Setting initial point in form of 'ndarray' as 'minimize' function requires it in form of 'ndarray'
+x3_start = np.array([30])
+# print(type(x))  -->  <class 'numpy.ndarray'>
+# print(x.shape)  -->  (1,)
+# Finding minimum of the function from starting point 'x_start = 30'
+# Using 'minimize' function from 'scipy.optimize' library
+y3_min = optimize.minimize(h_array, x3_start, method='BFGS')
+print(y3_min)  # fun = -5, iterations = 0
+print()
+
+# Finding minimum of non-smooth function with 'differential evolution' method
+# Searching in the range [1, 30]
+# Setting the range for searching in form of tuple inside list as function requires it
+x4_range = [(1, 30)]  # tuple inside list
+# Finding minimum of smooth function
+y4_min = optimize.differential_evolution(h_array, [(1, 30)])
+print(y4_min)  # fun = -11.0, iterations = 3
+print()
+
+# Plotting smooth and non-smooth functions with found points
+# Creating a figure with subplots
+figure, ax = plt.subplots(nrows=1, ncols=2)
+# ax is (1, 2) np array and to make it easier to read we use 'flatten' function
+# Or we can call each time ax[0, 0]
+ax0, ax1 = ax.flatten()
+
+# Preparing data for plotting smooth function
+# Creating list with numpy method 'arange'
+x0 = np.arange(1, 30, 0.1)
+y0 = f_array(x0)
+
+# Plotting smooth function in range [1, 30]
+# Giving title to the figure and axises
+ax0.set_title('Smooth function f(x)')
+ax0.set_xlabel('x')
+ax0.set_ylabel('y')
+# Showing dashed lines to the found minimums
+# By defining 'ymax' or 'xmax' we limit the end of the lines
+# 'ymax' or 'xmax' can be in range between 0 and 1
+ax0.axvline(x=4.1362, ymax=0.85, color='c', linestyle='--', linewidth=0.8)
+ax0.axhline(y=1.7452, xmax=0.14, color='c', linestyle='--', linewidth=0.8)
+ax0.axvline(x=25.8801, ymax=0.05, color='r', linestyle='--', linewidth=0.8)
+ax0.axhline(y=-11.8988, xmax=0.83, color='r', linestyle='--', linewidth=0.8)
+# Another way to draw the limited line is as following:
+# ax0.plot(np.arange(1, 10, 2), np.array([2]*5))  # arrays fo axises 'x' and 'y' has to have the same dimensions
+# Creating the figure for smooth function
+ax0.plot(x0, y0, 'b', 4.1362, 1.7452, 'co', 25.8801, -11.8988, 'ro')
+
+
+# Preparing data for plotting non-smooth function
+# Creating list with numpy method 'arange'
+x1 = np.arange(1, 30, 0.1)
+y1 = h_array(x1)
+
+# Plotting non-smooth function in range [1, 30]
+# Giving title to the figure and axises
+ax1.set_title('Non-smooth function h(int(f(x)))')
+ax1.set_xlabel('x')
+ax1.set_ylabel('y')
+# Showing dashed lines to the found minimum
+# By defining 'ymax' or 'xmax' we limit the end of the lines
+# 'ymax' or 'xmax' can be in range between 0 and 1
+ax1.axvline(x=25.0463, ymax=0.05, color='r', linestyle='--', linewidth=0.8)
+ax1.axhline(y=-11, xmax=0.85, color='r', linestyle='--', linewidth=0.8)
+ax1.axvline(x=30, ymax=0.43, color='c', linestyle='--', linewidth=0.8)
+ax1.axhline(y=-5, xmax=0.95, color='c', linestyle='--', linewidth=0.8)
+# Another way to draw the limited line is as following:
+# ax0.plot(np.arange(1, 10, 2), np.array([2]*5))  # arrays fo axises 'x' and 'y' has to have the same dimensions
+# Creating the figure for non-smooth function
+ax1.plot(x1, y1, 'b', 30, -5, 'co', 25.0463, -11.0, 'ro')
+
+
+# Function to make distance between figures
+plt.tight_layout()
+# Giving the name to the window with figures
+figure.canvas.set_window_title('Smooth and non-smooth functions to be optimized')
+# Showing result
+plt.show()