Leap Frog Optimizer Package - Lite Edition

• 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 and may be found on the website:

• 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.

This is the stripped down version of the package with minimal tools. It is written in pure Python to allow compatibility for the alpha versions until the full version can be released.

Installation

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

System requirements for installation:

• Python >= 3.6
• numpy
• scipy
• nose

Via pip

Lpfgopt may be installed with pip using the following command:

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

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

• Download the 'lite' branch 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 'lite' 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
  

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

$ 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)
subject to:
	g(x) <= 0
	bound[0][0] <= x[0] <= bound[0][1]
	bound[1][0] <= x[0] <= bound[1][1]
	...
	bound[n][0] <= x[0] <= bound[n][1]

where n is the number of decsision variables and bound
is a n X 2 list of lists or 2d numpy array with shape (n,2)

lpfgopt.minimize documentation

lpfgopt.minimize( fun, bounds, args=(), points=20, fconstraint=None, discrete=[], maxit=10000, tol=1e-5, seedval=None, pointset=None, callback=None )

General-use wrapper function to interface with the LeapFrog optimizer class. Contains the data and methods necessary to run a LeapFrog optimization. Accepts constraints, discrete variables and allows for a variety of options.

• Parameters:

  • fun : callable Objective function
  • bounds : array-like, shape (n, 2) Decision variable upper and lower bounds
  • args : iterable Other arguments to be passed into the function
  • points : int Point set size
  • fconstraint : callable Constraint function of the form g(x) <= 0
  • discrete : array-like List of indices that correspond to discrete variables. These variables will be constrained to integer values by truncating any randomly generated number (i.e. rounding down to the nearest integer absolute value)
  • maxit : int Maximum iterations
  • tol : float Convergence tolerance
  • seedval : int Random seed
  • pointset : array-like, shape (m, n) Starting point set
  • callback : callable Function to be called after each iteration

• Returns:

  • solution : dict A dictionary containing the results of the optimization. The members of the solution are listed below.

    • x : list The solution vector or the vector of decision variables that produced the lowest objective function value

    • success : bool Whether or not the optimizer exited successfully.

    • status : int Termination status of the optimizer. Its value depends on the underlying solver. Refer to message for details.

    • message : string Description of the cause of the termination.

    • fun: float The objective function value at 'x'

    • nfev : int The number of function evaluations of the objective function

    • nit : int The number of iterations performed

    • maxcv : float The maximum constraint violation evaluated during optimization

    • best : list The member of the population that had the lowest objective value in the point set having the form [f(x), x[0], x[1], ..., x[n-1]]

  • worst : list The member of the population that had the highest objective value in the point set having the form [f(x), x[0], x[1], ..., x[n-1]]

    • final_error : float The optimization convergence value upon termination

    • pointset : list, shape(m, n) The entire point set state upon termination having the form:

       [
       [f(x[0]),   x[0][0],   x[0][1],   ..., x[0][n-1]],
       [f(x[1]),   x[1][0],   x[1][1],   ..., x[1][n-1]],
       ...,
       [f(x[m-1]), x[m-1][0], x[m-1][1], ..., x[m-1][n-1]]
       ]
      

      where n is the number of decision variables and m is the number of points in the search population.

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

# 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: