landscapes 0.0.12

A Python library of standardized optimization test functions

Installation

There are a couple ways in which you can use this library. The first and probably the easiest is by using pip and PyPi:

pip install landscapes

You can also install directly from this git repo:

pip install git+https://github.com/nathanrooy/landscapes

Lastly, you can always clone/download this repo and use as is.

wget https://github.com/nathanrooy/landscapes/archive/master.zip
unzip master.zip
cd landscapes-master

Available functions from: single_objective

function name	method	dimensions
Ackley	ackley()	2
Ackley N.2	ackley_n2()	2
Adjiman	adjiman()	2
AMGM	amgm()	n
Bartels Conn	bartels_conn()	2
Bird	bird()	2
Beale	beale()	2
Bent Cigar	bent_cigar()	n
Bohachevsky N.1	bohachevsky_n1()	2
Bohachevsky N.2	bohachevsky_n2()	2
Bohachevsky N.3	bohachevsky_n3()	2
Booth	booth()	2
Branin	branin()	2
Brent	brent()	2
Brown	brown()	n
Bukin n6	bukin_n6()	2
3-Hump Camel	camel_hump_3()	2
6-Hump Camel	camel_hump_6()	2
Carrom Table	carrom_table()	2
Chichinadze	chichinadze()	2
Chung Reynolds	chung_reynolds()	n
Colville	colville()	4
Cosine Mixture	cosine_mixture()	n
Cross-in-Tray	cross_in_tray()	2
Csendes	csendes()	n
Cube	cube()	2
Damavandi	damavandi()	2
Deckkers-Aarts	deckkers_aarts()	2
Dixon & Price	dixon_price()	n
Drop Wave	drop_wave()	2
Easom	easom()	2
Eggholder	eggholder()	2
Exponential	exponential()	n
Freudenstein Roth	freudenstein_roth()	2
Goldstein–Price	goldstein_price()	2
Griewank	griewank()	n
Himmelblau	himmelblau()	2
Hölder table	holder_table()	2
Hosaki	hosaki()	2
Keane	keane()	2
Leon	leon()	2
Lévi function N.13	levi_n13()	2
Matyas	matyas()	2
Michalewicz	michalewicz	n
McCormick	mccormick()	2
Parsopoulos	parsopoulos()	2
Pen Holder	pen_holder()	2
Plateau	plateau()	n
Qing	qing()	n
Quartic	quartic()	n
Rastrigin	rastrigin()	n
Rotated Hyper-Ellipsoid	rotated_hyper_ellipsoid()	n
Rosenbrock	rosenbrock()	n
Salomon	salomon()	n
Schaffer N.2	schaffer_n2()	2
Schaffer N.4	schaffer_n4()	2
Schwefel	schwefel()	n
Sphere	sphere()	n
Step	step()	n
Styblinski–Tang	styblinski_tang()	n
Sum of Different Powers	sum_of_different_powers()	n
Sum of Squares	sum_of_squares()	n
Trid	trid()	n
Tripod	tripod()	2
Wolfe	wolfe()	3
Zakharov	zakharov()	n

Usage

As a simple example, let's use the Nelder-Mead method via SciPy to minimize the sphere function. We'll start off by importing the sphere function from Landscapes and the minimize method from SciPy.

>>> from landscapes.single_objective import sphere
>>> from scipy.optimize import minimize

Next, we'll call the minimize method using a starting location of [5,5].

>>> minimize(sphere, x0=[5,5], method='Nelder-Mead')

The output of which should look close to this:

 final_simplex: (array([[-3.33051318e-05, -1.93825710e-05],
       [ 4.24925225e-05,  1.37129516e-05],
       [ 3.09383247e-05, -4.04797876e-05]]), array([1.48491586e-09, 1.99365951e-09, 2.59579314e-09]))
           fun: 1.4849158640215086e-09
       message: 'Optimization terminated successfully.'
          nfev: 80
           nit: 44
        status: 0
       success: True
             x: array([-3.33051318e-05, -1.93825710e-05])

Function Reference - Single Objective

Ackley function

from landscapes.single_objective import ackley

global minimum	bounds	usage
f(x=0,y=0)=0	-5.12 <= x, y <= 5.12	ackley([x,y])

Beale function

from landscapes.single_objective import beale

global minimum	bounds	usage
f(x=3, y=0.5) = 0	-4.5 <= x, y <= 4.5	beale([x,y])

Booth function

from landscapes.single_objective import booth

global minimum	bounds	usage
f(x=1, y=3) = 0	-10 <= x, y <= 10	booth([x,y])

Bukin N.6 function

from landscapes.single_objective import bukin_n6

global minimum	bounds	usage
f(x=-10, y=1) = 0	-15 <= x <= -5 -3 <= y <= 3	bukin_n6([x,y])

Cross-in-tray function

from landscapes.single_objective import cross_in_tray

global minimum(s)	bounds	usage
f(x=1.34941, y=-1.34941) = -2.06261 f(x=1.34941, y=1.34941) = -2.06261 f(x=-1.34941, y=1.34941) = -2.06261 f(x=-1.34941, y=-1.34941) = -2.06261	-10 <= x, y <= 10	cross_in_tray([x,y])

Easom function

from landscapes.single_objective import easom

global minimum	bounds	usage
f(x=pi, y=pi) = -1	-100 <= x, y <= 100	easom([x,y])

Eggholder function

from landscapes.single_objective import eggholder

global minimum	bounds	usage
f(x=512, y=404.2319) = -959.6407	-512 <= x, y <= 512	eggholder([x,y])

Goldstein–Price function

from landscapes.single_objective import goldstein_price

global minimum	bounds	usage
f(x=0, y=-1) = 3	-2 <= x, y <= 2	goldstein_price([x,y])

Himmelblau's function

from landscapes.single_objective import himmelblau

global minimum(s)	bounds	usage
f(x=3.0, y=2.0) = 0.0 f(x=-2.805118, y=3.131312) = 0.0 f(x=-3.779310, y=-3.283186) = 0.0 f(x=3.584428, y=-1.848126) = 0.0	-5 <= x, y <= 5	himmelblau([x,y])

Hölder table function

from landscapes.single_objective import holder_table

global minimum(s)	bounds	usage
f(x=8.05502, y=9.66459) = -19.2085 f(x=-8.05502, y=9.66459) = -19.2085 f(x=8.05502, y=-9.66459) = -19.2085 f(x=-8.05502, y=-9.66459) = -19.2085	-10 <= x, y <= 10	holder_table([x,y])

Lévi function N.13

from landscapes.single_objective import levi_n13

global minimum	bounds	usage
f(x=1, y=1) = 0	-10 <= x, y <= 10	levi_n13([x,y])

Matyas function

from landscapes.single_objective import matyas

global minimum	bounds	usage
f(x=0, y=0) = 0	-10 <= x, y <= 10	matyas([x,y])

McCormick function

from landscapes.single_objective import mccormick

global minimum	bounds	usage
f(x=-0.54719, y=-1.54719) = -1.9133	-1.5 <= x <= 4 -3 <= y <= 4	mccormick([x,y])

Rastrigin function

from landscapes.single_objective import rastrigin

global minimum	bounds	usage
f([0,...,0]) = 0	-5.12 <= x_i <= 5.12	rastrigin([x_1,...,x_n])

Rosenbrock function

from landscapes.single_objective import rosenbrock

global minimum	bounds	usage
f([1,...,1]) = 0	-inf <= x_i <= inf	rosenbrock([x_1,...,x_n])

Schaffer function N.2

from landscapes.single_objective import schaffer_n2

global minimum	bounds	usage
f(x=0, y=0) = 0	-100 <= x, y <= 100	schaffer_n2([x,y])

Schaffer function N.4

from landscapes.single_objective import schaffer_n4

global minimum	bounds	usage
f(x=0, y=1.25313) = 0.292579 f(x=0, y=-1.25313) = 0.292579	-100 <= x, y <= 100	schaffer_n4([x,y])

Sphere function

from landscapes.single_objective import sphere

global minimum	bounds	usage
f([0,...,0]) = 0	-inf <= x_i <= inf	sphere([x_1,...x_n])

Styblinski–Tang function

from landscapes.single_objective import styblinski_tang

global minimum	bounds	usage
-39.16617n < f([-2.903534,...,-2.903534]) < -39.16616n	-5 <= x_i <= 5	styblinski_tang([x_1,...x_n])

Three-hump camel function

from landscapes.single_objective import camel_hump_3

global minimum	bounds	usage
-f(x=0, y=0) = 0	-5 <= x_i <= 5	three_hump_camel([x,y])

Travelling salesman problem (TSP)

from landscapes.single_objective import tsp

There are several ways to use the TSP function within Landscapes, all of which involve specifying a list of tsp stops, and a distance function.

Example 1: Multi-dimensional list of points using Euclidean distance function

from landscapes.single_objective import tsp
from scipy.spatial import distance

np.random.seed(10)
pts = np.random.rand(5,3)

which will yield a list of three-dimensional points:

array([[0.77132064, 0.02075195, 0.63364823],
       [0.74880388, 0.49850701, 0.22479665],
       [0.19806286, 0.76053071, 0.16911084],
       [0.08833981, 0.68535982, 0.95339335],
       [0.00394827, 0.51219226, 0.81262096]])

Then, initialize the tsp function:

tsp_cost = tsp(distance.euclidean, close_loop=True).dist

To calculate the total travel distance, simply call the function with the list of points:

tsp_cost(pts)
>>> 3.2043803044101096

The flag close_loop simply specifies whether the distance between the first and last points should be calculated.

Example 2: Specifying points using Latitude and Longitude

Insead of multi-dimensional points in space, let's specify a list of locations based on longitude and latitude then calculate the distances using the inverse Vincenty's formulae which is available in the spatial package [here].

First let's import our Vincenty based distance function and wrap it for easier use.

from spatial import vincenty_inverse as vi

def vi_tsp(p1, p2):
    return vi(p1, p2).mi() # output distance in miles

Next, let's specify some locations. Here are some breweries in Cincinnati. Each row represents a [longitude, latitude].

pts = [
    [-84.508661, 39.110187],
    [-84.520021, 39.117219],
    [-84.514938, 39.113937],
    [-84.517401, 39.111322],
    [-84.476906, 39.128957]]

Again, initialize the tsp function:

tsp_cost = tsp(vi_tsp, close_loop=True).dist

And finally, calculate the travel distance:

tsp_cost(pts)
>>> 5.993331331465468

Example #3: Geospatial distances on a graph

In Example #2 we used Vincenty's inverse formulae which calculates the distance between two longitude and latitude pairs "as the crow flies". That's great for some situations, but in a city where we're limited by streets and sidewalks, it's a little less useful. Instead, what we want is the actual distance if we were going to walk or bike. This is the network distance and is only slighly more complex, but involves some additinal libraries.

First, import the dependencies:

import osmnx as ox
import networkx as nx
import pandas as pd

Next, load the brewery locations (available here) and prepare the Open Street Map (OSM) network graph.

pts_df = pd.read_csv('brewery_locations.csv')

# determine bounds for osm network
lats = locs_df['lat'].values
lngs = locs_df['lng'].values

bbox = [
    max(lats) + 0.1,
    min(lats) - 0.1,
    max(lngs) + 0.1,
    min(lngs) - 0.1]

# download osm street network
G = ox.graph_from_bbox(bbox[0], bbox[1], bbox[2], bbox[3], network_type='drive')

Downloading the osm graph might take a bit depending on internet speed. Next, let's create a new cost function that takes in two brewery names and returns the network distance in meters.

def osm_dist(n0, n1):
    p0 = pts_df[pts_df['name']==n0][['lat','lng']].values[0]
    p1 = pts_df[pts_df['name']==n1][['lat','lng']].values[0]

    p0_node = ox.get_nearest_node(G, p0)
    p1_node = ox.get_nearest_node(G, p1)

    dist_m = nx.shortest_path_length(G, p0_node, p1_node, weight='length')
    return dist_m

Again, specify the tsp cost function:

tsp_cost = tsp(osm_dist, close_loop=True).dist

And to get the network distance:

tsp_cost(locs_df['name'].values)
>>> 75950.73399999998

This translates to roughly 47 miles.