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Question

Minimize the function E(x) = X4- 3 x3-15x -4 =0,using simulated annealing method. Take an initial random guess x1 between 0 and 10. Set the value of Temperature T= 100. Calculate E(x1). Choose another random guess x2. Calculate E(x2). Calculate the probability P : exp(- [E(x2)-E(x2)]/T). If P> 1, accept x2, otherwise choose another random guess. Continue 5 iterations. Then reduce the temperature to 90 and repeat the procedure. Then reduce the temperature to 80, 70, 60, 50, 40,30,20, and 10. Determine the optimum value of x (which will minimize E(x). [Submit your computer code and outputs after first 5 iterations for each temperature).

Answer 1

To solve the minimization problem using the simulated annealing method, you need to iterate through different temperatures and update the current guess based on the acceptance probability. Here's a Python code snippet that implements the simulated annealing algorithm for your function:

import random

import math

def E(x):

return x**4 - 3*x**3 - 15*x - 4

def acceptance_probability(delta_E, temperature):

if delta_E < 0:

return 1.0

return math.exp(-delta_E / temperature)

def minimize_simulated_annealing():

temperature = 100

current_guess = random.uniform(0, 10)

current_energy = E(current_guess)

print("Initial guess:", current_guess)

print("Initial energy:", current_energy)

for temp in range(100, 0, -10):

print("\nTemperature:", temp)

temperature = temp

for iteration in range(1, 6):

new_guess = random.uniform(0, 10)

new_energy = E(new_guess)

delta_energy = new_energy - current_energy

acceptance_prob = acceptance_probability(delta_energy, temperature)

print("Iteration:", iteration)

print("Current guess:", current_guess)

print("Current energy:", current_energy)

print("New guess:", new_guess)

print("New energy:", new_energy)

print("Delta energy:", delta_energy)

print("Acceptance probability:", acceptance_prob)

if acceptance_prob > random.uniform(0, 1):

current_guess = new_guess

current_energy = new_energy

print("Accepted new guess!")

else:

print("Rejected new guess!")

print("Optimum value of x:", current_guess)

minimize_simulated_annealing()

This code performs five iterations for each temperature value from 100 to 10, reducing the temperature by 10 after each iteration. The outputs for the first five iterations at each temperature are printed. You can run this code to see the computed results and adjust it according to your needs.