import numpy as np

from scipy.optimize import minimize

# Objective function

def objective$($x$)$:

x$1,$ x$2,$ x$3,$ x$4,$ x$5=$ x

return np$.$exp$($x$1*$ x$2*$ x$3*$ x$4*$ x$5)-0\mathrm{.}5*($x$1**3+$ x$2**3+1)**2$

# Constraints

def constraint$1($x$)$:

return x$[0]**2+$ x$[1]**2+$ x$[2]**2+$ x$[3]**2+$ x$[4]**2-10$

def constraint$2($x$)$:

return x$[1]*$ x$[2]-5*$ x$[3]*$ x$[4]$

def constraint$3($x$)$:

return x$[0]**3+$ x$[1]**3+1$

# Algorithm $18\mathrm{.}1$

def SQP$\_$algorithm$($objective$,$ constraints, x$0)$:

x $=$ x$0$

while True:

# Step $1$: Solve the Quadratic Programming Subproblem

result $=$ minimize$($objective$,$ x$,$ constraints$=$constraints, method$=\text{'}$SLSQP$\text{'})$

# Step $2$: Check for Convergence

if np$.$linalg.norm$($result$.$x $-$ x$)<1$e$-6$:

break

# Step $3$: Update x

x $=$ result.x

return result

# Initial guess and constraints

x$0=$ np$.$array$([-1\mathrm{.}71,1\mathrm{.}59,1\mathrm{.}82,-0\mathrm{.}763,-0\mathrm{.}763])$

con$1=\{\text{'}$type$\text{'}$: $\text{'}$eq$\text{'},$ 'fun': constraint$1\}$

con$2=\{\text{'}$type$\text{'}$: $\text{'}$eq$\text{'},$ 'fun': constraint$2\}$

con$3=\{\text{'}$type$\text{'}$: $\text{'}$eq$\text{'},$ 'fun': constraint$3\}$

constraints $=[$con$1,$ con$2,$ con$3]$

# Solve the optimization problem using Algorithm $18\mathrm{.}1$

result $=$ SQP$\_$algorithm$($objective$,$ constraints, x$0)$

# Print the solution

print$("$Optimal solution:"$)$

print$("$x$*=",$ result.x$)$