Leapfrog Optimizer Package

• Author : Mark Redd
• Email : redddogjr@gmail.com
• GitHub : https://github.com/flythereddflagg
• Website : http://www.r3eda.com/

About

This package is based the Leapfrogging Optimization Algorithm published by Dr. R. Russell Rhinehart. The following publications explain the technique:

• Rhinehart, R. R., M. Su, and U. Manimegalai-Sridhar, “Leapfrogging and Synoptic Leapfrogging: a new optimization approach”, Computers & Chemical Engineering, Vol. 40, 11 May 2012, pp. 67-81.

• Manimegalai-Sridhar, U., A. Govindarajan, and R. R. Rhinehart, “Improved Initialization of Players in Leapfrogging Optimization”, Computers & Chemical Engineering, Vol. 60, 2014, 426-429.

• Rhinehart, R. R., “Convergence Criterion in Optimilsation of Stochastic Processes”, Computers & Chemical Engineering, Vol. 68, 4 Sept 2014, pp 1-6.

Installation

You can install via pip or using the setup.py script in the source code. Instructions are shown below.

Minimum system Requirements for Installation

• Python >= 3.6

Recommended System Requirements for Installation

• Windows or Linux
• x86 processor with a 64-bit architecture
• Additional Python Packages:
  • scipy
  • numpy
  • pytest

Via pip

The lpfgopt package may be installed with pip using the following command:

$ pip install lpfgopt # You may need root privileges or the --user tag

If you wish to install locally with pip you may do the following:

• Download and unzip the archive or clone it with git.
• Open the main directory where setup.py is located and run the following command:

  $ pip install .
  

Via setup.py

• Download the master branch and unzip the archive or clone it with git.
• Open the main directory where "setup.py" is located and run the following command:

  $ python setup.py install     # You may need root priviliges or use the --user tag
  

Test the Installation

The software should be installed correctly. You may validate the installation by executing the following commands in a terminal:

$ python
>>> import lpfgopt
>>> lpfgopt.__version__
'X.X.X'
>>> lpfgopt.minimize(lambda x: x[0]**2 + 10, [[-10, 10]])['x']
[<approximately 0.0>]
>>>

If the above commands produce the output congratulations! You have successfully installed the package!

Usage

Use the lpfgopt.minimize function to solve optimization problems of the form:

$minimize$ $f(x)$ $s.t.:$

• $g(x) \le 0$

• $bound_{1,1} \le x[1] \le bound_{1,2}$

  $bound_{2,1} \le x[2] \le bound_{2,2}$

  $...$

  $bound_{n,1} \le x[n] \le bound_{n,2}$

where $n$ is the number of decision variables and $bound$ is an array-like with shape $(n, 2)$. $bound$ may be a list of lists or a numpy ndarray.

Example Usage

The following is a simple optimization where the minimum value of the following equation is found:

• $f(x) = x^2+y^2$

• Subject to:

  $g(x) = -x^2 - y + 10 \le 0$ or g(x) = -x^2 - y + 10 <= 0

  $$x, y \in [-5, 5]$$

# test_lpfgopt.py
from lpfgopt import minimize
import matplotlib.pyplot as plt

# set up the objective funciton, 
# constraint fuction and bounds
f = lambda x: sum([i**2 for i in x])
g = lambda x: -x[0]**2 + 10 - x[1] 
bounds = [[-5,5] for i in range(2)]

# run the optimization
sol = minimize(f, bounds, fconstraint=g)['x']
print(f"Solution is: {sol}")

# plot the results on a contour plot
gg = lambda x: -x**2 + 10 # for plotting purposes

plt.figure(figsize=(8,8))
x, y = np.linspace(-5,5,1000), np.linspace(-5,5,1000)
X, Y = np.meshgrid(x,y)
Z = f([X,Y])

plt.contourf(X,Y,Z)
plt.plot(x, gg(x), "r", label="constraint")
plt.plot(*sol, 'x', 
         markersize=14, 
         markeredgewidth=4, 
         color="lime", 
         label="optimum")
plt.ylim(-5,5)
plt.xlim(-5,5)
plt.legend()
plt.show()

This code will produce the following output:

Solution is: [-3.0958051486911997, 0.4159905027317925]

As well as a plot that should look similar to the following image: