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x = m.Array(m.Var,4,value=1,lb=1,ub=5) x1,x2,x3,x4 = x

1. change initial values

x2.value = 5; x3.value = 5

m.Equation(x1**2+x2**2+x3**2+x4**2==40) m.Minimize(x1*x4*(x1+x2+x3)+x3)

print('x: ', x) print('Objective: ',m.options.OBJFCNVAL)

m.solve()

x1.lower = 1 x2.lower = 1 x3.lower = 1 x4.lower = 1

x1.upper = 5 x2.upper = 5 x3.upper = 5 x4.upper = 5

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(:source lang=python:)

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(:toggle hide scipy button show="Show SciPy Solution":) (:div id=scipy:)

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Method #3: GEKKO Solution

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(:toggle hide gekko button show="Show GEKKO Solution":) (:div id=gekko:) from gekko import GEKKO import numpy as np

1. Initialize Model

m = GEKKO()

1. help(m)
2. define parameter

eq = m.Param(value=40)

1. initialize variables

x1,x2,x3,x4 = [m.Var() for i in range(4)]

1. initial values

x1.value = 1 x2.value = 5 x3.value = 5 x4.value = 1

1. lower bounds

x1.lb = 1 x2.lb = 1 x3.lb = 1 x4.lb = 1

1. upper bounds

x1.ub = 5 x2.ub = 5 x3.ub = 5 x4.ub = 5

1. Equations

m.Equation(x1*x2*x3*x4>=25) m.Equation(x1**2+x2**2+x3**2+x4**2==eq)

1. Objective

m.Obj(x1*x4*(x1+x2+x3)+x3)

1. Set global options

m.options.IMODE = 3 #steady state optimization

1. Solve simulation

m.solve(remote=True)

1. Results

print('') print('Results') print('x1: ' + str(x1.value)) print('x2: ' + str(x2.value)) print('x3: ' + str(x3.value)) print('x4: ' + str(x4.value)) (:divend:)

print('Initial Objective: ' + str(objective(x0)))

print('Final Objective: ' + str(objective(x)))

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Method #1: APM Python

Method #2: SciPy Optimize Minimize

(:source lang=python:) import numpy as np from scipy.optimize import minimize

def objective(x):

    return x[0]*x[3]*(x[0]+x[1]+x[2])+x[2]

def constraint1(x):

    return x[0]*x[1]*x[2]*x[3]-25.0

def constraint2(x):

    sum_eq = 40.0
    for i in range(4):
        sum_eq = sum_eq - x[i]**2
    return sum_eq

1. initial guesses

n = 4 x0 = np.zeros(n) x0[0] = 1.0 x0[1] = 5.0 x0[2] = 5.0 x0[3] = 1.0

1. show initial objective

print('Initial SSE Objective: ' + str(objective(x0)))

1. optimize

b = (1.0,5.0) bnds = (b, b, b, b) con1 = {'type': 'ineq', 'fun': constraint1} con2 = {'type': 'eq', 'fun': constraint2} cons = ([con1,con2]) solution = minimize(objective,x0,method='SLSQP', bounds=bnds,constraints=cons) x = solution.x

1. show final objective

print('Final SSE Objective: ' + str(objective(x)))

1. print solution

print('Solution') print('x1 = ' + str(x[0])) print('x2 = ' + str(x[1])) print('x3 = ' + str(x[2])) print('x4 = ' + str(x[3])) (:sourceend:)

$$\min x_1 x_4 \left(x_1 + x_2 + x_3\right) + x_3$$ $$\mathrm{s.t.} \quad x_1 x_2 x_3 x_4 \ge 25$$ $$x_1^2 + x_2^2 + x_3^2 + x_4^2 = 40$$ $$1\le x_1, x_2, x_3, x_4 \le 5$$ $$x_0 = (1,5,5,1)$$

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Mathematical optimization problems may include equality constraints (e.g. =), inequality constraints (e.g. <, <=, >, >=), objective functions, algebraic equations, differential equations, continuous variables, discrete or integer variables, etc. One example of an optimization problem from a benchmark test set is the Hock Schittkowski problem #71.

(:title Optimization with Python:) (:keywords Optimization, Nonlinear Programming, Python, nonlinear, optimization, engineering optimization, university course:) (:description Optimization with Python - Problem-Solving Techniques for Chemical Engineers at Brigham Young University:)

Optimization deals with selecting the best option among a number of possible choices that are feasible or don't violate constraints. Python can be used to optimize parameters in a model to best fit data, increase profitability of a potential engineering design, or meet some other type of objective that can be described mathematically with variables and equations.

Mathematical optimization problems may include equality constraints (e.g. =), inequality constraints (e.g. <, <=, >, >=), objective functions, algebraic equations, differential equations, continuous variables, discrete or integer variables, etc. One example of an optimization problem from a benchmark test set is the Hock Schittkowski problem #71.

This problem has a nonlinear objective that the optimizer attempts to minimize. The variable values at the optimal solution are subject to (s.t.) both equality (=40) and inequality (>25) constraints. The product of the four variables must be greater than 25 while the sum of squares of the variables must also equal 40. In addition, all variables must be between 1 and 5 and the initial guess is x₁ = 1, x₂ = 5, x₃ = 5, and x₄ = 1.

• Download HS71 Example Problem in Python
  

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This tutorial can also be completed with nonlinear programming optimizers that are available with the Excel Solver and MATLAB Optimization Toolbox. Click on the appropriate link for additional information and source code.

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