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Constrained optimization using scipy

Constrained optimization using scipy

Randall Romero Aguilar, PhD

This demo is based on the original Matlab demo accompanying the Computational Economics and Finance 2001 textbook by Mario Miranda and Paul Fackler.

Original (Matlab) CompEcon file: demopt08.m

Running this file requires the Python version of CompEcon. This can be installed with pip by running

!pip install compecon --upgrade

Last updated: 2021-Oct-01

About

The problem is

\[\begin{equation*} \max\{-x_0^2 - (x_1-1)^2 - 3x_0 + 2\} \end{equation*}\]

subject to

\[\begin{align*} 4x_0 + x_1 &\leq 0.5\\ x_0^2 + x_0x_1 &\leq 2.0\\ x_0 &\geq 0 \\ x_1 &\geq 0 \end{align*}\]

Using scipy

The scipy.optimize.minimize function minimizes functions subject to equality constraints, inequality constraints, and bounds on the choice variables.

import numpy as np
from scipy.optimize import minimize

np.set_printoptions(precision=4,suppress=True)

  • First, we define the objective function, changing its sign so we can minimize it

def f(x):
    return x[0]**2 + (x[1]-1)**2 + 3*x[0] - 2

  • Second, we specify the inequality constraints using a tuple of two dictionaries (one per constraint), writing each of them in the form \(g_i(x) \geq 0\), that is

\[\begin{align*} 0.5 - 4x_0 - x_1 &\geq 0\\ 2.0 - x_0^2 - x_0x_1 &\geq 0 \end{align*}\]
cons = ({'type': 'ineq', 'fun': lambda x: 0.5 - 4*x[0] - x[1]},
       {'type': 'ineq', 'fun': lambda x: 2.0 - x[0]**2 - x[0]*x[1]})

  • Third, we specify the bounds on \(x\):

\[\begin{align*} 0 &\leq x_0 \leq \infty\\ 0 &\leq x_1 \leq \infty \end{align*}\]
bnds = ((0, None), (0, None))

  • Finally, we minimize the problem, using the SLSQP method, starting from \(x=[0,1]\)

x0 = [0.0, 1.0]
res = minimize(f, x0, method='SLSQP', bounds=bnds, constraints=cons)
print(res)

     fun: -1.7499999999999876
     jac: array([ 3., -1.])
 message: 'Optimization terminated successfully'
    nfev: 10
     nit: 3
    njev: 3
  status: 0
 success: True
       x: array([0. , 0.5])

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