Hello,

I need help with the following task in PYTHON. The optimization algorithms that need to be

tested are:

Random Search

Hill Climb

Iterated local search

Simulated Annealing

Objective: Find the maximum value of $f(x,y,z),$ where

$f(x,y,z)=e^{sin(40z)}+sin(60cos(z))+e^{sin(50x)}$

$+sin(60e^y)+sin(70sin(x))+sin(sin(80y))$

$-sin(10(x+y))+\frac{x^2+y^2+z^2}{100}$

Constraints: The solution must be subject to the $($hard$)$ constraints:

$0x,y,z5,$ and $,x,y,$zinR

You should explain the approach taken, submitting all programming code that is used $-$

provide comments to code, or a description of the code, where appropriate to demonstrate

you understand the code used.

Equation in Python:

math. exp$($ math $*sin(40**z))+$ math $*sin(60**$ math $*cos(z))+$ math $*exp($ math $*sin(50**x))$

$+$ math. $sin(60**math*exp(y))+$ math $sin(70**math*sin(x))+$ math $*sin($ math $*sin(80**y))$

$-$ math $sin(10**(x+y))+\frac{x****2+y****2+z****2}{100}$

Examples of algorithms in Python:

#Random Search

import random

import numpy

import math

def f$($x$)$:

f $=-$x$**6+28*$x$**5-307*$x$**4+1660*$x$**3-4564*$x$**2+5872*$x $+2688$ #Change this for each question

return f

xbest $=$ random.uniform$(0,10)$

fbest $=$ f$($xbest$)$

steps $=100$

for i in range$(1,$steps$)$:

xnew $=$ random.uniform$(0,10)$ #new random solution

fnew $=$ f$($xnew$)$

if fnew $>$ fbest:

xbest $=$ xnew

fbest $=$ fnew

print $(\text{'}$xbest$\text{'},$ xbest, 'fbest', fbest,$)$

# Perform a hill$-$climb

def f$($x$)$:

f $=-$x$**6+28*$x$**5-307*$x$**4+1660*$x$**3-4564*$x$**2+5872*$x $+2688$ #Change this for each question

return f

xbest $=$ random.uniform$(0,10)$

fbest $=$ f$($xbest$)$

steps $=100$

for i in range$(1,$steps$)$:

xnew $=$ xbest $+$ random.gauss$(0,0\mathrm{.}1)$ #Hill climb step

fnew $=$ f$($xnew$)$

if fnew $>$ fbest:

xbest $=$ xnew

fbest $=$ fnew

print$(\text{'}$xbest$\text{'},$ xbest, 'fbest', fbest$)$

# Perform a Iterated Local Search $($ILS$)$

# The function to optimise

def f$($x$)$:

f $=-$x$**6+28*$x$**5-307*$x$**4+1660*$x$**3-4564*$x$**2+5872*$x $+2688$ #Change this for each question

return f

# Create a random initial solution, with $0$ fbest: # if new solution is better than best solution, update best solution.

xbest $=$ xnew

fbest $=$ fnew

# Continue ILS Loop $-$ Check best solution against overall optimal solution

if fbest $>$ fopt: # if new solution is better than best solution, update best solution.

xopt $=$ xbest

fopt $=$ fbest

# Carry out a big step

xbest $=$ xopt $+$ random.gauss$(0,1)$ #step size here should be small, $1$ here

### print best solution

print$(\text{'}$xopt$\text{'},$ xopt, 'fopt', fopt$)$

# Simulated Annealing

def f$($x$)$:

f $=-$x$**6+28*$x$**5-307*$x$**4+1660*$x$**3-4564*$x$**2+5872*$x $+2688$ #Change this for each question

return f

xbest $=$ random.uniform$(0,10)$

fbest $=$ f$($xbest$)$

steps $=200$

for i in range$(1,$steps$)$:

T$=5*(1-$i$/$steps$)$ #Tempertaure function

xnew $=$ xbest $+$ random.gauss$(0,0\mathrm{.}1)$

fnew $=$ f$($xnew$)$

if fnew $>$ fbest:

xbest $=$ xnew

fbest $=$ fnew

elif random.random$()$ math.exp$(($fnew$-$fbest$)/$T$)$: #Accept worse solutions?

xbest $=$ xnew

fbest $=$ fnew

print $(\text{'}$xbest$\text{'},$ xbest, 'fbest', fbest,$)$