Searching extreme points of polyhedron

Question

In my Uni, my scientific professor asked me to make some researches about the extreme points of polyhedrals. And I did them. I found that there is still no code in public for searching extreme points for polyhedral with n dimensions (n - x's), but polyhedrons are everywhere (CV, game theories, etc.). I wrote a function for this task and made a python library (also there are matrix and array combination maker functions).

All I want is to make this code more optimal and compatible for all python versions (I have some troubles that some times happen when I install it by "pip install lin", but other times no). I want to make the life of people easier and make it more comfortable.

I am asking for you to test this function on your computer and write if you have any bugs, fails or thoughts on how to make it better. I am open to constructive criticism and non-constructive too (it will help to understand if somebody needs it or that is just a waste of time).

All the examples, instructions and code on my GitHub: https://github.com/r4ndompuff/polyhedral_set

import numpy as np
import itertools as it
import math
import re

def permutation(m,n):
    return math.factorial(n)/(math.factorial(n-m)*math.factorial(m))

def matrix_combinations(matr,n):
    timed = list(map(list, it.combinations(matr, n)))
    for i in range(n):
        timed[i][i][i] = np.asscalar(timed[i][i][i])
    all = np.array(list(timed))
    return all

def array_combinations(arr,n):
    timed = list(map(list, it.combinations(arr, n)))
    for i in range(n):
        timed[i][i] = np.asscalar(timed[i][i])
    all = np.array(list(timed))
    return all

def check_extreme(matr, arr, x, sym_comb, m):
    sym_comb = sym_comb.replace(']', '')
    sym_comb = sym_comb.replace('[', '')
    sym_comb = re.split("[ ,]", sym_comb)
    for i in range(m):
        td_answer = sum(matr[i]*x)
        if sym_comb[i] == '>':
            if td_answer <= arr[i]:
                return 0
        elif sym_comb[i] == '>=':
            if td_answer < arr[i]:
                return 0
        elif sym_comb[i] == '<':
            if td_answer >= arr[i]:
                return 0
        elif sym_comb[i] == '<=':
            if td_answer > arr[i]:
                return 0
        elif sym_comb[i] == '=':
            if td_answer != arr[i]:
                return 0
        elif sym_comb[i] == '!=':
            if td_answer == arr[i]:
                return 0
        else:
            return 0
    return 1

def extreme_points(m,n,A,b,sym_comb):
    # Input
    A = np.array(A).reshape(m,n)
    b = np.array(b).reshape(m,1)
    # Proccess
    ans_comb = np.zeros((1,n))
    arr_comb = array_combinations(b,n)
    matr_comb = matrix_combinations(A,n)
    for i in range(int(permutation(n,m))):
        if np.linalg.det(matr_comb[i]) != 0:
            x = np.linalg.solve(matr_comb[i],arr_comb[i])
            ans_comb = np.vstack([ans_comb,x])
    ans_comb = np.delete(ans_comb, (0), axis=0)
    j = 0
    for i in range(len(ans_comb)):
        if check_extreme(A, b, ans_comb[j], sym_comb, m):
            ans_comb = ans_comb
            j = j + 1
        else:
            ans_comb = np.delete(ans_comb, (j), axis=0)
    # Output
    return ans_comb

And I am uploading some more tests. https://i.sstatic.net/2eRl4.jpg

Can you provide some more information on what exactly is going on, from a math standpoint? Is this actually a polyhedron, or a polytope? In how many dimensions? By "extreme", what do you mean? Euclidean norm from (the origin, an arbitrary point)? — Reinderien
– Reinderien, Commented Apr 21, 2019 at 23:21
"there is still no code in public for searching extreme points for polyhedral with n dimensions" - I can nearly guarantee that that isn't the case. — Reinderien
– Reinderien, Commented Apr 21, 2019 at 23:22
@Reinderien I am still in the process of deciphering the question. I have a math background, and I share the mother tongue with OP. My impression is that by extremal points OP means the vertices of a simplex where a certain linear form (defined by A, b, and the condition) achieves an extremum. I could be wrong. — vnp
– vnp, Commented Apr 22, 2019 at 0:32
@vnp That's kind of what I guessed, and if that's the case, linear programming is a quite well-established field already - with some stuff built right into scipy. — Reinderien
– Reinderien, Commented Apr 22, 2019 at 0:33

Reinderien · Accepted Answer · 2019-04-22 16:03:53Z

I created a rudimentary pull request to your GitHub repo. I won't show all of the content here except for the main file:

import numpy as np
import itertools as it
from math import factorial
import re


def permutation(m, n):
    return factorial(n) / (factorial(n - m) * factorial(m))


def matrix_combinations(matr, n):
    timed = list(map(list, it.combinations(matr, n)))
    return np.array(list(timed))


def array_combinations(arr, n):
    timed = list(map(list, it.combinations(arr, n)))
    return np.array(list(timed))


def check_extreme(matr, arr, x, sym_comb, m):
    sym_comb = sym_comb.replace(']', '')
    sym_comb = sym_comb.replace('[', '')
    sym_comb = re.split("[ ,]", sym_comb)
    for i in range(int(m)):
        td_answer = sum(matr[i] * x)
        if sym_comb[i] == '>':
            if td_answer <= arr[i]:
                return 0
        elif sym_comb[i] == '>=':
            if td_answer < arr[i]:
                return 0
        elif sym_comb[i] == '<':
            if td_answer >= arr[i]:
                return 0
        elif sym_comb[i] == '<=':
            if td_answer > arr[i]:
                return 0
        elif sym_comb[i] == '=':
            if td_answer != arr[i]:
                return 0
        elif sym_comb[i] == '!=':
            if td_answer == arr[i]:
                return 0
        else:
            return 0
    return 1


def extreme_points(A, b, sym_comb):
    # Input
    A = np.array(A)
    b = np.array(b)
    m, n = A.shape
    # Process
    ans_comb = np.zeros((1, n))
    arr_comb = array_combinations(b, n)
    matr_comb = matrix_combinations(A, n)
    for i in range(int(permutation(n, m))):
        if np.linalg.det(matr_comb[i]) != 0:
            x = np.linalg.solve(np.array(matr_comb[i], dtype='float'),
                                np.array(arr_comb[i], dtype='float'))
            ans_comb = np.vstack([ans_comb, x])
    ans_comb = np.delete(ans_comb, 0, axis=0)
    j = 0
    for i in range(len(ans_comb)):
        if check_extreme(A, b, ans_comb[j], sym_comb, m):
            ans_comb = ans_comb
            j += 1
        else:
            ans_comb = np.delete(ans_comb, j, axis=0)
    # Output
    return ans_comb

Notable changes:

Do a direct import of factorial
Don't call asscalar, since it's both unneeded and deprecated
Don't call a variable all, since that shadows a Python built-in
Don't need to explicitly pass array dimensions, nor do you need to reshape the arrays
Drop redundant parens around some expressions
Use += where applicable
Fix up almost all PEP8 issues, except for your capital letter A, which is fine in context

This doesn't solve the bigger issue that you should replace 99% of this with a call to scipy. I'll do that separately (I suspect that @vnp is, as well).

Thank you! In meantime I made it work with any NumPy version now (about loop error you report). — Andrey Lovyagin
– Andrey Lovyagin, Commented Apr 22, 2019 at 7:57
Sorry, but I can't find the same function in scipy. Can you provide me the name of function? — Andrey Lovyagin
– Andrey Lovyagin, Commented Apr 22, 2019 at 8:17
@AndreyLovyagin docs.scipy.org/doc/scipy/reference/generated/… — Reinderien
– Reinderien, Commented Apr 22, 2019 at 16:18
Thank you very much for all that you have done! Also, I checked scipy.optimize.linprog() function and I can't really understand how it can replace much of the code. I will learn and read more about it, but on first sight, it looks like it can't solve non-equational system (I mean not '=', but any sign, e.g. '>='), that is why I can't see how I can find extreme points by it. — Andrey Lovyagin
– Andrey Lovyagin, Commented Apr 22, 2019 at 20:39

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