# solves *bounded* LPs of the form: # max cx # sub to: Ax <= b from sympy import * from itertools import combinations # enumerates all the vertices of {x | Ax <= b} def enumeratevertices(A, b): m, n = A.rows, A.cols for rowlist in combinations(range(m), n): Ap = A.extract(rowlist, range(n)) bp = b.extract(rowlist, [0]) if Ap.det() != 0: xp = Ap.LUsolve(bp) d = A * xp - b feasible = True for i in range(m): if d[i] > 0: feasible = False if feasible: yield xp # finds the optimum using vertex enumeration def findoptimum(A, b, c): m, n = A.rows, A.cols bestvalue, bestvertex = None, None for vertex in enumeratevertices(A, b): if bestvalue is None or (vertex.T*c)[0] > bestvalue: bestvalue = (vertex.T * c)[0] bestvertex = vertex return bestvertex def solve(A, b, c): x = findoptimum(A, b, c) if not x: print 'LP is infeasible' else: print 'Vertex', x.T, 'is optimal' print 'Optimal value is', c.T*x if __name__ == '__main__': A = Matrix([[-10, -6, -9, -10], [ 8, -6, -5, -5], [ -7, -1, -9, 3], [ -1, -4, 5, 10], [ 1, 2, 0, 10], [ 2, -9, 3, -8], [ -8, -1, -8, 1], [ 7, -10, 4, -4], [-10, 2, 5, 8], [ -7, 9, 4, -4], [ -1, 0, 0, 0], [ 0, -1, 0, 0], [ 0, 0, -1, 0], [ 0, 0, 0, -1]]) b = Matrix([9, 7, 3, 4, 8, 0, 3, 2, 4, 8, 0, 0, 0, 0]) c = Matrix([2, -2, -3, 8]) solve(A, b, c)