def solve_fw_tap(A, b, times, caps, I, K, maxiter=100, tol = 0.001):
h = matrix(0., (I*K, 1))
G = spdiag([-1.]*I*K)
block = build_block_matrix(I, K)
reverse_block = reverse_block_matrix(I, K)
#start iteration
#flows = matrix(0.0, (I*K,1))
#start with a normal lp
times_expanded = reverse_block * times
print 'computing first feasible flow'
flows = solvers.lp(times_expanded, G, h, A = A, b = b, solver='mosek', options={"mosek":{iparam.log:0}})['x']
for k in range(maxiter):
print 'iter', k
#compute gradient
sum_flows = block * flows
print 'flows', max(sum_flows),min(sum_flows),max(flows), min(flows)
Df = vdf_derivative(times, caps, sum_flows)
Df = reverse_block * Df
print 'Df', max(Df), min(Df)
#solve affineminimization subproblem -> xd
lpsol = solvers.lp(Df, G, h, A = A, b = b, solver='mosek', options={"mosek":{iparam.log:0}})
xd = lpsol['x']
print lpsol['status']
if lpsol['status'] != 'optimal':
return {'flows':flows, 'grad':Df, 'G':G, 'h':h}
#update
bound = (Df.T * (flows - xd))[0]
step = (2. / (k + 2.)) * (xd - flows)
print 'diff', np.sqrt(step.T * step)[0], bound
if abs(bound) < tol:
print "finished after " + str(k) + " iterations"
break
flows = flows + step
return flows
def solve_LP_problem(self):
(f_coef_matrix, f_column_vector) = self.build_function_coef_matrix_and_column_vector()
(d_coef_matrix, d_column_vector) = self.build_derivative_coef_matrix_and_column_vector()
# Solve the LP problem by combining constraints for both function and derivative info.
objective_function_vector = matrix(list(itertools.repeat(1.0, self.no_vars)))
coef_matrix = sparse([f_coef_matrix, d_coef_matrix])
column_vector = matrix([f_column_vector, d_column_vector])
min_sol = solvers.lp(objective_function_vector, coef_matrix, column_vector)
is_consistent = min_sol['x'] is not None
# Print the LP problem for debugging purposes.
if self.verbose:
self.display_LP_problem(coef_matrix, column_vector)
if is_consistent:
self.min_heights = np.array(min_sol['x']).reshape(self.no_points_per_axis)
print np.around(self.min_heights, decimals=2)
# Since consistency has been established, solve the converse LP problem to get the
# maximal bounding surface.
max_sol = solvers.lp(-objective_function_vector, coef_matrix, column_vector)
self.max_heights = np.array(max_sol['x']).reshape(self.no_points_per_axis)
print np.around(self.max_heights, decimals=2)
if self.plot_surfaces:
self.plot_3D_objects_for_2D_case()
else:
print 'No witness for consistency found.'
return is_consistent
def ty_solver(data, l, w_toll, full=False):
"""Solves the block (t,y) of the toll pricing model
Parameters
----------
data: Aeq, beq, ffdelays, coefs
l: linkflows
w_toll: weight on the toll collected
full: if True, return the whole solution
"""
Aeq, beq, ffdelays, coefs = data
delays = compute_delays(l, ffdelays, coefs)
n = len(l)
p = Aeq.size[1]/n
m = Aeq.size[0]/p
c = matrix([(1.0+w_toll)*l, -beq])
I = spdiag([1.0]*n)
G1 = matrix([-I]*(p+1))
G2 = matrix([Aeq.T, matrix(0.0, (n,p*m))])
G = matrix([[G1],[G2]])
h = [delays]*p
h.append(matrix(0.0, (n,1)))
h = matrix(h)
if full: return solvers.lp(c,G,h)['x']
return solvers.lp(c,G,h)['x'][range(n)]
def test_options(self):
from cvxopt import glpk, solvers
c,G,h,A,b = self._prob_data
glpk.options = {'msg_lev' : 'GLP_MSG_OFF'}
sol1 = glpk.lp(c,G,h)
self.assertTrue(sol1[0]=='optimal')
sol2 = glpk.lp(c,G,h,A,b)
self.assertTrue(sol2[0]=='optimal')
sol3 = glpk.lp(c,G,h,options={'msg_lev' : 'GLP_MSG_ON'})
self.assertTrue(sol3[0]=='optimal')
sol4 = glpk.lp(c,G,h,A,b,options={'msg_lev' : 'GLP_MSG_ERR'})
self.assertTrue(sol4[0]=='optimal')
sol5 = solvers.lp(c,G,h,solver='glpk',options={'glpk':{'msg_lev' : 'GLP_MSG_ON'}})
self.assertTrue(sol5['status']=='optimal')
sol1 = glpk.ilp(c,G,h,None,None,set(),set([0,1]))
self.assertTrue(sol1[0]=='optimal')
sol2 = glpk.ilp(c,G,h,A,b,set([0,1]),set())
self.assertTrue(sol2[0]=='optimal')
sol3 = glpk.ilp(c,G,h,None,None,set(),set([0,1]),options={'msg_lev' : 'GLP_MSG_ALL'})
self.assertTrue(sol3[0]=='optimal')
sol4 = glpk.ilp(c,G,h,A,b,set(),set([0]),options={'msg_lev' : 'GLP_MSG_ALL'})
self.assertTrue(sol4[0]=='optimal')
solvers.options['glpk'] = {'msg_lev' : 'GLP_MSG_ON'}
sol5 = solvers.lp(c,G,h,solver='glpk')
self.assertTrue(sol5['status']=='optimal')
def test_options(self):
from cvxopt import matrix, msk, solvers
msk.options = {msk.mosek.iparam.log: 0}
c = matrix([-4., -5.])
G = matrix([[2., 1., -1., 0.], [1., 2., 0., -1.]])
h = matrix([3., 3., 0., 0.])
msk.lp(c, G, h)
msk.lp(c, G, h, options={msk.mosek.iparam.log: 1})
solvers.lp(c, G, h, solver='mosek', options={'mosek':{msk.mosek.iparam.log: 1}})
def cvxopt_solver(G, h, A, b, c, n):
# cvxopt doesn't allow redundant constraints in the linear program Ax = b,
# so we need to do some preprocessing to find and remove any linearly
# dependent rows in the augmented matrix [A | b].
#
# First we do Gaussian elimination to put the augmented matrix into row
# echelon (reduced row echelon form is not necessary). Since b comes as a
# numpy array (that is, a row vector), we need to convert it to a numpy
# matrix before transposing it (that is, to a column vector).
b = np.mat(b).T
A_b = np.hstack((A, b))
A_b, permutation = to_row_echelon(np.hstack((A, b)))
# Next, we apply the inverse of the permutation applied to compute the row
# echelon form.
P = dictionary_to_permutation(permutation)
A_b = P.I * A_b
# Trim any rows that are all zeros. The call to np.any returns an array of
# Booleans that correspond to whether a row in A_b is all zeros. Indexing
# A_b by an array of Booleans acts as a selector. We need to use np.asarray
# in order for indexing to work, since it expects a row vector instead of a
# column vector.
A_b = A_b[np.any(np.asarray(A_b) != 0, axis=1)]
# Split the augmented matrix back into a matrix and a vector.
A, b = A_b[:, :-1], A_b[:, -1]
# Apply the linear programming solver; cvxopt requires that these are all
# of a special type of cvx-specific matrix.
G, h, A, b, c = (cvx_matrix(M) for M in (G, h, A, b, c))
solution = cvx_solvers.lp(c, G, h, A, b)
if solution['status'] == 'optimal':
return True, solution['x']
# TODO status could be 'unknown' here, but we're currently ignoring that
return False, None
def CVXOPT_LP_Solver(p, solverName):
#os.close(1); os.close(2)
if solverName == 'native_CVXOPT_LP_Solver': solverName = None
cvxopt_solvers.options['maxiters'] = p.maxIter
cvxopt_solvers.options['feastol'] = p.contol
cvxopt_solvers.options['abstol'] = p.ftol
if p.iprint <= 0:
cvxopt_solvers.options['show_progress'] = False
cvxopt_solvers.options['LPX_K_MSGLEV'] = 0
cvxopt_solvers.options['MSK_IPAR_LOG'] = 0
xBounds2Matrix(p)
#WholeRepr2LinConst(p)
# CVXOPT have some problems with x0 so currently I decided to avoid using the one
#if p.x0.size>0 and p.x0.flatten()[0] != None and all(isfinite(p.x0)):
# sol= cvxopt_solvers.solvers.lp(Matrix(p.f), Matrix(p.A), Matrix(p.b), Matrix(p.Aeq), Matrix(p.beq), solverName)
#else:
if (len(p.intVars)>0 or len(p.binVars)>0) and solverName == 'glpk':
from cvxopt.glpk import ilp
c = Matrix(p.f)
A, b = Matrix(p.Aeq), Matrix(p.beq)
G, h = Matrix(p.A), Matrix(p.b)
if A is None:
A = matrix(0.0, (0, p.n))
b = matrix(0.0,(0,1))
if G is None:
G = matrix(0.0, (0, p.n))
h = matrix(0.0,(0,1))
(status, x) = ilp(c, G, h, A, b, set(p.intVars), B=set(p.binVars))
if status == 'optimal': p.istop = SOLVED_WITH_UNIMPLEMENTED_OR_UNKNOWN_REASON
elif status == 'maxiters exceeded': p.istop = IS_MAX_ITER_REACHED
elif status == 'time limit exceeded': p.istop = IS_MAX_TIME_REACHED
elif status == 'unknown': p.istop = UNDEFINED
else: p.istop = FAILED_WITH_UNIMPLEMENTED_OR_UNKNOWN_REASON
if x is None:
p.xf = nan*ones(p.n)
else:
p.xf = array(x).flatten()#w/o flatten it yields incorrect result in ff!
p.ff = sum(p.dotmult(p.f, p.xf))
p.msg = status
else:
# if len(p.b) != 0:
# s0 = matrix(p.b - dot(p.A, p.x0))
# else:
# s0 = matrix()
# primalstart = {'x': matrix(p.x0), 's': s0}
# sol = cvxopt_solvers.lp(Matrix(p.f), Matrix(p.A), Matrix(p.b), Matrix(p.Aeq), Matrix(p.beq), solverName, primalstart)
sol = cvxopt_solvers.lp(Matrix(p.f), Matrix(p.A), Matrix(p.b), Matrix(p.Aeq), Matrix(p.beq), solverName)
p.msg = sol['status']
if p.msg == 'optimal' : p.istop = SOLVED_WITH_UNIMPLEMENTED_OR_UNKNOWN_REASON
else: p.istop = -100
if sol['x'] is not None:
p.xf = asarray(sol['x']).flatten()
# ! don't involve p.ff - it can be different because of goal and additional constant from FuncDesigner
p.duals = concatenate((asarray(sol['y']).flatten(), asarray(sol['z']).flatten()))
else:
p.ff = nan
p.xf = nan*ones(p.n)
def l1Min(A,b):
"""
Solve min ||x||_1 s.t. Ax = b using CVXOPT.
Inputs:
A -- numpy array of shape m x n
b -- numpy array of shape m
Returns:
x -- numpy array of shape n
"""
m, n = A.shape
A1 = np.zeros((m,2*n))
A1[:,n:] = A
c = np.zeros(2*n)
c[:n] = 1
h = np.zeros(2*n)
G = np.zeros((2*n,2*n))
G[:n,:n] = -np.eye(n)
G[:n,n:] = np.eye(n)
G[n:,:n] = -np.eye(n)
G[n:,n:] = -np.eye(n)
c = matrix(c)
G = matrix(G)
h = matrix(h)
A = matrix(A1)
b = matrix(b)
sol = solvers.lp(c, G, h, A, b)
return np.array(sol['x'])[n:].flatten()
def generate_program(alpha): assert(alpha < 1) t1 = alpha**2 t2 = (1 - alpha)**2 t3 = 2*alpha*(1 - alpha) tmp_list_A = [ \ [1., 2., 1., 0., 0., 0.], \ [0., 1., 0., 2., 1., 1.], \ [0., 0., 1., 0., 2., 1.], \ [t1, t1 + t3, t2, t3, t2, 0.]] \ # [0., t1, t1, t3, t2 + t3, t2], ] tmp_list_B = [1., 1., 1., 1 - alpha] #, alpha] A = matrix(tmp_list_A).T b = matrix(tmp_list_B) tmp_G_list = [ \ [-1., 0., 0., 0., 0., 0.], \ [0., -1., 0., 0., 0., 0.], \ [0., 0., -1., 0., 0., 0.], \ [0., 0., 0., -1., 0., 0.], \ [0., 0., 0., 0., -1., 0.], \ [0., 0., 0., 0., 0., -1.]] tmp_h_list = [-0.001, 0., 0., 0., 0., -0.001] G = matrix(tmp_G_list).T h = matrix(tmp_h_list) c = matrix([1., 1., 1., 1., 1., 1.]) sol = solvers.lp(c, G, h, A, b) return (G, h, A, b, c, sol)
def Norm_inf1 (X,W):
# X, W are two matrix of shape respectively f*n et r*n
f,n = X.size
r = W.size[0]
print f,n,r
P = matrix(W).trans()
F = matrix(1.0,(f,r))
onesn = matrix(1.0,(n,1))
Idn = spmatrix(1.0, range(n),range(n))
Idr = spmatrix(1.0, range(r),range(r))
Zrn = spmatrix(0,[r-1],[n-1])
Zr = spmatrix(0,[r-1],[0])
#Zn = spmatrix(0,[n-1],[0])
A = sparse([ [P,-P,-Idr], [-Idn,-Idn,Zrn] ])
for i in range(f):
V = X[i,:].trans()
C = matrix([[V,-V,Zr]])
e = matrix([ [Zr, onesn] ])
solution = solvers.lp(e,A,C)['x']
F[i,:] = solution[range(r)].trans()
return F
def prob2():
"""Solve the transportation problem by converting all equality constraints
into inequality constraints.
Returns (in order):
The optimizer (sol['x'])
The optimal value (sol['primal objective'])
"""
# minimize this function
c = matrix([4.0, 7.0, 6.0, 8.0, 8.0, 9.0])
# constraints
G = matrix(
[
[1.0, -1.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[1.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 1.0, -1.0, 0.0, 0.0, 1.0, -1.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 1.0, -1.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 1.0, -1.0, 1.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 1.0, -1.0, 0.0, 0.0, 1.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0],
]
)
# with respect to these values
h = matrix([7.0, -7.0, 2.0, -2.0, 4.0, -4.0, 5.0, -5.0, 8.0, -8.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0])
# compute the solution to the modified constraints
sol = solvers.lp(c, G, h)
return sol["x"], sol["primal objective"]
def __solve(self):
sol = solvers.lp(self.__generate_c(), self.__generate_A(), self.__generate_b())
vector = array(sol['x'])
vector = around(vector, decimals=0)
return vector
def l1Min(A, b):
"""Calculate the solution to the optimization problem
minimize ||x||_1
subject to Ax = b
Return only the solution x (not any slack variable), as a flat NumPy array.
Parameters:
A ((m,n) ndarray)
b ((m, ) ndarray)
Returns:
x ((n, ) ndarray): The solution to the minimization problem.
"""
m, n = A.shape
b = matrix(b.astype(float))
c = matrix(np.hstack((np.ones(n), np.zeros(n))))
G = np.hstack((-np.eye(n), np.eye(n)))
Gi = np.hstack((-np.eye(n), -np.eye(n)))
G = matrix(np.vstack((G,Gi)))
h = matrix(np.zeros(2*n))
A = matrix(np.hstack((np.zeros((m,n)), A)))
sol = solvers.lp(c, G, h, A, b)
return sol['x'][n:]
def generate_lp(): tmp_list_A = [ #1, 2, 3, 4, 5, 6, 7, d, e, f, a, a', b, b', c, c', g, g', h, h' [1., 0., 1., 0., 0., 1., 1., 0., 0., 0., 0., 1., 0., 1., 1., 0., 1., 0., 0., 1.], #1 [0., 1., 0., 1., 1., 0., 0., 1., 1., 1., 1., 0., 1., 0., 0., 1., 0., 1., 1., 0.], #2 [0., 0., 1., 0., 0., 1., 1., 1., 1., 0., 0., 0., 0., 1., 1., 0., 1., 0., 0., 0.], #3 [0., 0., 0., 1., 1., 0., 0., 1., 1., 1., 0., 0., 1., 0., 0., 1., 0., 1., 0., 0.], #4 [0., 0., 0., 0., 0., 1., 0., 0., 0., 1., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0.], #5 [0., 0., 0., 0., 1., 0., 0., 0., 1., 1., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0.], #6 [0., 0., 0., 0., 0., 0., 1., 0., 1., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0.], #7 [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 1., 0., 0., 0., 0., 0., 0., 0., 0.], #A [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 1., 0., 0., 0., 0., 0., 0.], #B [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 1., 0., 0., 0., 0.], #C [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 1., 0., 0.], #G [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 1.]] #H tmp_list_b = [5., 8., 5., 6., 2., 3., 2., 2., 2., 2., 2., 2.] tmp_list_G = [[-1. if i == j else 0. for i in range(20)] for j in range(20)] tmp_list_h = [0. for i in range(20)] G = matrix(tmp_list_G).T h = matrix(tmp_list_h) A = matrix(tmp_list_A).T b = matrix(tmp_list_b) c = matrix([-1., -1., -1., -1., -1., -1., -1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]) sol = solvers.lp(c, G, h, A, b) return (G, h, A, b, c, sol)
def pfOptimizer(longTickers, shortTickers, Coef, StockBeta, BETA_BOUND, WEIGHT_BOUND):
longTickers["bWeight"] = 1 # binary weights
shortTickers["bWeight"] = -1
pfTickers = (
pd.concat((shortTickers[["ticker"]], longTickers[["ticker"]]), axis=0).sort("ticker").reset_index(drop=True)
)
# sens = Coef[['ticker', 'Mkt-RF', 'SMB', 'HML', 'UMD']].merge(pfTickers).rename(columns={'Mkt-RF': 'beta'})
# control = matrix([1, 0.2, 0.2, 0.5])
# scores = matrix(sens[['beta', 'SMB', 'HML', 'UMD']].as_matrix()) * control
betas = Coef[["ticker", "Mkt-RF"]].merge(pfTickers).rename(columns={"Mkt-RF": "beta"}).reset_index(drop=True)
mBeta = matrix(betas["beta"])
mCqaBeta = matrix(StockBeta.merge(pfTickers)["cqaBeta"])
longIndex = matrix(pfTickers.merge(longTickers, how="left").fillna(0)["bWeight"])
mLongIndex = np.diag(pfTickers.merge(longTickers, how="left").fillna(0)["bWeight"])
mLongIndex = matrix(mLongIndex[np.logical_or.reduce([np.sum(mLongIndex, 1) > 0.5])]).trans()
# mLongIndex = matrix(np.diag(tickers.merge(longTickers, how='left').fillna(0)['bWeight']))
shortIndex = -matrix(pfTickers.merge(shortTickers, how="left").fillna(0)["bWeight"])
mShortIndex = -np.diag(pfTickers.merge(shortTickers, how="left").fillna(0)["bWeight"])
mShortIndex = matrix(mShortIndex[np.logical_or.reduce([np.sum(mShortIndex, 1) > 0.5])]).trans()
# mShortIndex = matrix(np.diag(pfTickers.merge(shortTickers, how='left').fillna(0)['bWeight']))
# wTickers = functions.iniWeights(pfTickers, shortTickers, longTickers) # initial weights
wStart = matrix(functions.iniWeights(pfTickers, longTickers, shortTickers)["weight"])
N = pfTickers.shape[0]
id = spmatrix(1.0, range(N), range(N))
wBounds = matrix(np.ones((N, 1)) * WEIGHT_BOUND)
longBounds = matrix(np.zeros((shortTickers.shape[0], 1)))
# longBounds = matrix(np.ones((shortTickers.shape[0], 1)) * 0.002)
shortBounds = matrix(np.zeros((longTickers.shape[0], 1)))
# shortBounds = matrix(np.ones((longTickers.shape[0], 1)) * (-0.005))
A = matrix(
[
[mCqaBeta],
[-mCqaBeta],
[longIndex],
[-longIndex],
[shortIndex],
[-shortIndex],
[id],
[-id],
[-mLongIndex],
[mShortIndex],
]
).trans()
b = matrix([BETA_BOUND, BETA_BOUND, 1, -0.98, -0.98, 1, wBounds, wBounds, longBounds, shortBounds])
# A = matrix([ [longIndex], [-longIndex],
# [shortIndex], [-shortIndex],
# [id], [-id],
# [-mLongIndex], [mShortIndex]]).trans()
# b = matrix([ 1, -0.98, -0.98, 1, wBounds, wBounds, longBounds, shortBounds])
# scores = mBeta
# sol = solvers.lp(-scores, A, b)
sol = solvers.lp(-mBeta, A, b)
w_res = sol["x"]
wTickers = pfTickers
wTickers["weight"] = w_res
return wTickers
def problem_1():
"""
This function provides the solution for problem 1 in the file:
'lab3 - linaer-programming.pdf'
Inputs:
None
Returns:
A dictionary containing the solution and a lot of other info as defined
in cvxopt.solvers.lp
"""
c = matrix([-1.0, -2.0, -3.0, -4.0, -5.0])
G = matrix([[5.0, 4.0, 3.0, 2.0, 1.0,],
[1.0, -2.0, 3.0, -4.0, 5.0],
[-1.0, 2.0, -3.0, 4.0, -5.0],
[-1.0, 0, 0 ,0 ,0],
[0, -1.0, 0, 0, 0],
[0, 0, -1.0, 0, 0],
[0, 0, 0, -1.0, 0],
[0, 0, 0, 0, -1.0]])
G = G.T
h = matrix([30.0, 20.0, 10.0, 0., 0., 0., 0., 0.])
solution = solvers.lp(c, G, h)
return solution
def opt_queue(mu, K, l, t):
q = [0] * K
N = np.zeros([K, K])
ucb_avg = np.zeros([K, K])
queue = np.empty((0, K), int)
for i in range(K):
mui = np.amax(mu[i, :])
lam = l[i]
q[i] = stat_sample(lam, mui, 1, 100)
queue = np.append(queue, np.array([q]), axis=0)
c, G, h = schedule(K, l, mu)
sol = solvers.lp(c, G, h)
solx = sol["x"]
hp = []
for i in range(1, len(solx)):
hp = hp + [solx[i]]
result = BVN(hp, K)
prob = [pit[1] for pit in result]
values = range(len(prob))
for s in range(1, t):
index = weighted_values(values, prob, 1)
# (q,N,ucb_avg) = one_schedule(result[index][0],mu,K,q,N,ucb_avg)
one_schedule(result[index][0], mu, K, q, N, ucb_avg, l)
queue = np.append(queue, np.array([q]), axis=0)
return queue
def cheby_center(C,D,b):
'''Calculates the chebyshev center for polytope
C x + D y <= b
Input:
`C, D, b`: Polytope parameters
Output:
`x_0, y_0`: The chebyshev centra
`boolean`: True if a point could be found, False otherwise'''
d = C.shape[1]
k = D.shape[1]
A = np.hstack([C,D])
dim = np.shape(A)[1]
c = -np.r_[np.zeros(dim),1]
norm2 = np.sqrt(np.sum(A*A, axis=1))
G = np.c_[A, norm2]
solvers.options['show_progress']=False
solvers.options['LPX_K_MSGLEV'] = 0
sol = solvers.lp(matrix(c), matrix(G), matrix(b), None, None, lp_solver)
if sol['status'] == "optimal":
opt = np.array(sol['x'][0:-1]).flatten()
return opt[range(0,d)], opt[range(d,d+k)], True
else:
return np.zeros(d), np.zeros(k), False
def linprog(*args, **kwargs):
"""
min c^T*x
s.t. Gx <= h
Ax = b
args: c, G, h, A, b
"""
verbose = kwargs.get('verbose', False)
# Save settings and set verbosity
old_settings = _apply_options({'show_progress': verbose,
'LPX_K_MSGLEV': int(verbose)})
# Optimize
results = lp(*args, solver='glpk')
# Restore settings
_apply_options(old_settings)
# Check return status
status = results['status']
if not status == 'optimal':
from sys import stderr
print >> stderr, ('Warning: termination of lp with status: %s'
% status)
# Convert back to NumPy array
# and return solution
xstar = np.array(results['x'])
return xstar, results['primal objective']
def problem_10(X, y):
"""
This function implements the minimization of ||X*beta - y|| for the one
norm.
Inputs:
X: The matrix X that defines the characterizing equations for the model.
y: The vector so that X*beta = y
Returns:
beta: A NumPy array representing the vector that minimizes the objective
function.
Notes:
This function uses the cvxopt.solvers.lp routine to get the answer.
"""
X = np.mat(X, dtype = float)
y = np.array(y, dtype = float)
m,n = X.shape
B = np.eye(m)
G = np.vstack((np.hstack((X, -B)), np.hstack((-X, -B))))
c = np.hstack((np.zeros(n), np.ones(m)))
h = np.hstack((y, -y))
c = matrix(c)
h = matrix(h)
G = matrix(G)
solution = solvers.lp(c, G, h)
return np.array(solution['x'])[:n]
def prob6():
"""Solve the allocation model problem in 'ForestData.npy'.
Note that the first three rows of the data correspond to the first
analysis area, the second group of three rows correspond to the second
analysis area, and so on.
Returns (in order):
The optimizer x (ndarray)
The optimal value (sol['primal objective']*-1000)
"""
data = np.load("ForestData.npy")
c = matrix(data[:, 3] * -1)
A = la.block_diag(*[[1.0, 1.0, 1.0] for _ in xrange(7)])
b = data[::3, 1].copy()
G = np.vstack((-data[:, 4], -data[:, 5], -data[:, 6], -np.eye(21))) # flip the inequality signs
h = np.hstack(([-40000.0, -5.0, -70.0 * 788.0], np.zeros(21))) # flip the inequality signs
c = matrix(c)
A = matrix(A)
b = matrix(b)
G = matrix(G)
h = matrix(h)
sol = solvers.lp(c, G, h, A, b)
return np.ravel(sol["x"]), sol["primal objective"] * -1000.0
def quadprog(H, c, A, b):
x = np.matrix(solvers.lp(matrix(np.zeros(c.shape)), matrix(-A), matrix(-b))['x'])
# x = scipy.optimize.linprog(np.ones(c.shape).T[0], -np.array(A), -np.array(b).T[0])['x']
# x = np.matrix(x).T
workset = np.array(A * x <= b).T[0]
max_iter = 200
for k in xrange(max_iter):
Aeq = A[np.where(workset == True)]
g_k = H * x + c
p_k, lambda_k = qpsub(H, g_k, Aeq, np.matlib.zeros((Aeq.shape[0], 1)))
if np.linalg.norm(p_k) <= 1e-9:
if np.min(lambda_k) > 0:
break
else:
pos = np.argmin(lambda_k)
for i in xrange(b.size):
if workset[i] and np.sum(workset[:i]) == pos:
workset[i] = False
break
else:
alpha = 1.0
pos = -1
for i in xrange(b.size):
if not workset[i] and (A[i] * p_k)[0, 0] < 0:
now = np.abs((b[i] - A[i] * x) / (A[i] * p_k))[0, 0]
if now < alpha:
alpha = now
pos = i
x += alpha * p_k
if pos != -1:
workset[pos] = True
return x
def static_stat_sample(lam,mu,K,num):
c,G,h = schedule(K,lam,mu)
sol = solvers.lp(c,G,h)
l = sol['x']
M = soltobvn(l,K)
result = BVN(M,K)
q = np.array([0]*K)
N = np.zeros([K,K])
ucb_avg = np.zeros([K,K])
prob = [pit[1] for pit in result]
values = range(len(prob))
for s in range(num):
index = weighted_values(values,prob,1)
one_schedule(result[index][0],mu,K,q,N,ucb_avg,lam)
return q
def unique_equalityset(C,D,b,af,bf,abs_tol=1e-7,verbose=0):
'''Return the equality set E such that
P_E = {x | af x = bf} intersection P
where P is the polytope C x + D y < b
The inequalities have to be satisfied with equality everywhere on
the face defined by af and bf.'''
if D is not None:
A = np.hstack([C,D])
a = np.hstack([af, np.zeros(D.shape[1]) ])
else:
A = C
a = af
E = []
for i in range(A.shape[0]):
A_i = np.array(A[i,:])
b_i = b[i]
sol = solvers.lp(matrix(A_i), matrix(A) , matrix(b), matrix(a).T, matrix(bf), lp_solver)
if sol['status'] != "optimal":
raise Exception("unique_equalityset: LP returned status " + str(sol['status']))
if np.abs(sol['primal objective'] - b_i) < abs_tol:
# Constraint is active everywhere
E.append(i)
if len(E) == 0:
raise Exception("unique_equalityset: empty E")
return np.array(E)
def problem_2():
"""
This function provides the solution for problem 2 in the file:
'lab3 - linaer-programming.pdf'
Inputs:
None
Returns:
A dictionary containing the solution and a lot of other info as defined
in cvxopt.solvers.lp
"""
c = matrix([-6., -8., -5., -9.])
G = matrix([[2., 1., 1., 3.],
[1., 3., 1., 2.],
[-1., 0., 0., 0.],
[0., -1.0, 0., 0.],
[0., 0., -1.0, 0.],
[0., 0., 0., -1.0]])
G = G.T
h = matrix([5., 3., 0., 0., 0., 0.])
A = matrix([1., 1., 1.,0.])
A = A.T
b = matrix([1.])
solution = solvers.lp(c, G, h, A, b)
return solution
def main():
'''
Main function
'''
ore = EveOnlineManufacturingJob(DATA_ACCESS_OBJECT)
reproc_mat_list = ore.get_mineral_matrix_adjusted(sec_status_low_limit=0.9,
fclt_base_yield=0.54,
rprcs_skill_lvl=5,
rprcs_eff_skill_lvl=5,
mtrl_spcfc_prcs_skill_lvl=5,
implant_bonus=0)
# for simplyfication we filter ore list
reproc_mat_list_filtered = {}
reproc_mat_list_filtered[DATA_ACCESS_OBJECT.get_inv_type(type_name='Veldspar')['type_id']] = reproc_mat_list[DATA_ACCESS_OBJECT.get_inv_type(type_name='Veldspar')['type_id']]
# reproc_mat_list_filtered[DATA_ACCESS_OBJECT.get_inv_item(type_name='Plagioclase')['type_id']] = reproc_mat_list[DATA_ACCESS_OBJECT.get_inv_item(type_name='Plagioclase')['type_id']]
# reproc_mat_list_filtered[DATA_ACCESS_OBJECT.get_inv_item(type_name='Scordite')['type_id']] = reproc_mat_list[DATA_ACCESS_OBJECT.get_inv_item(type_name='Scordite')['type_id']]
reproc_mat_list_filtered[DATA_ACCESS_OBJECT.get_inv_type(type_name='Pyroxeres')['type_id']] = reproc_mat_list[DATA_ACCESS_OBJECT.get_inv_type(type_name='Pyroxeres')['type_id']]
reproc_mat_list = reproc_mat_list_filtered
# define mineral amounts we want to get refining the ores
mineral_amounts_desired={}
mineral_amounts_desired[DATA_ACCESS_OBJECT.get_inv_type(type_name='Tritanium')['type_id']] = 200
mineral_amounts_desired[DATA_ACCESS_OBJECT.get_inv_type(type_name='Nocxium')['type_id']] = 1
# mineral_amounts_desired[DATA_ACCESS_OBJECT.get_inv_item(type_name='Pyerite')['type_id']] = 160
# mineral_amounts_desired[DATA_ACCESS_OBJECT.get_inv_item(type_name='Mexallon')['type_id']] = 80
# define variables for building matrices
list_of_mineral_matrices = []
mineral_amounts_desired_matrix = []
ore_quantity_matrix = []
# build matrices for linear programming
for k_ore,v_minerals in reproc_mat_list.iteritems():
min_quantity_list = [-float(quant) for quant in v_minerals.values()]
list_of_mineral_matrices.append(min_quantity_list)
ore_quantity_matrix.append(float(1.0))
min_id_list = v_minerals.keys()
for min_id in min_id_list:
if (min_id not in mineral_amounts_desired.keys()):
mineral_amounts_desired_matrix.append(float(0.0))
else:
mineral_amounts_desired_matrix.append(-float(mineral_amounts_desired[min_id]))
A = matrix(list_of_mineral_matrices)
b = matrix(mineral_amounts_desired_matrix)
c = matrix(ore_quantity_matrix)
printing.options['dformat'] = '%.1f'
printing.options['width'] = -1
print A
print b
print c
sol=solvers.lp(c,A,b)
print(sol['x'])
def pfOptimizer7(longTickers, shortTickersHigh, shortTickersLow,
Coef, Res, CarhartDaily, StockBeta,
BETA_BOUND, WEIGHT_BOUND, back_date, start_date):
shortTickers = shortTickersHigh.append(shortTickersLow).sort('ticker').reset_index(drop=True)
longTickers['bWeight'] = 1 # binary weights
shortTickers['bWeight'] = -1
shortTickersLow['bWeight'] = -1
pfTickers = pd.concat((shortTickers[['ticker']], longTickers[['ticker']]),
axis=0).sort('ticker').reset_index(drop=True)
sens = Coef[['ticker', 'Mkt-RF', 'SMB', 'HML', 'UMD']].merge(pfTickers)
mBeta = matrix(sens['Mkt-RF'])
mBeta_smb = matrix(sens['SMB'])
mBeta_hml = matrix(sens['HML'])
mBeta_umd = matrix(sens['UMD'])
mCqaBeta = matrix(StockBeta.merge(pfTickers)['cqaBeta'])
longIndex = matrix(pfTickers.merge(longTickers, how='left').fillna(0)['bWeight'])
mLongIndex = np.diag(pfTickers.merge(longTickers, how='left').fillna(0)['bWeight'])
mLongIndex = matrix(mLongIndex[np.logical_or.reduce([np.sum(mLongIndex, 1) > 0.5])]).trans()
shortIndex = -matrix(pfTickers.merge(shortTickers, how='left').fillna(0)['bWeight'])
mShortIndex = -np.diag(pfTickers.merge(shortTickers, how='left').fillna(0)['bWeight'])
mShortIndex = matrix(mShortIndex[np.logical_or.reduce([np.sum(mShortIndex, 1) > 0.5])]).trans()
shortLowIndex = -matrix(pfTickers.merge(shortTickersLow, how='left').fillna(0)['bWeight'])
N = pfTickers.shape[0]
id = spmatrix(1.0, range(N), range(N))
wBounds = matrix(np.ones((N, 1)) * WEIGHT_BOUND)
longBounds = matrix(np.zeros((shortTickers.shape[0], 1)))
shortBounds = matrix(np.zeros((longTickers.shape[0], 1)))
# total_cov = matrix(functions.get_cov(pfTickers, Coef, Res, CarhartDaily, back_date, start_date))
# q = matrix(np.zeros((pfTickers.shape[0], 1)))
q = -mBeta_smb - mBeta_hml
G = matrix([[mBeta], [-mBeta],
# [mBeta_smb], [-mBeta_smb],
# [mBeta_hml], [-mBeta_hml],
[mBeta_umd], [-mBeta_umd],
[mCqaBeta], [-mCqaBeta],
[longIndex], [-longIndex],
[shortIndex], [-shortIndex],
[-shortLowIndex],
[id], [-id],
[-mLongIndex], [mShortIndex]]).trans()
opt_comp = pd.DataFrame()
h = matrix([0.1, 0.2,
# 0.1, 0.1,
# 0.1, 0.1,
0.1, 0.1,
BETA_BOUND, BETA_BOUND,
1, -0.98,
-0.98, 1,
0.3,
wBounds, wBounds,
longBounds, shortBounds])
sol = solvers.lp(q, G, h)
w_final = sol['x']
wTickers = pfTickers
wTickers['weight'] = w_final
return (opt_comp, wTickers)
def pfOptimizer_sector(longTickers, shortTickers,
Coef, StockBeta, industry_data,
BETA_BOUND, WEIGHT_BOUND, SECTOR_WEIGHT):
longTickers['bWeight'] = 1 # binary weights
shortTickers['bWeight'] = -1
pfTickers = pd.concat((shortTickers[['ticker']], longTickers[['ticker']]), axis=0).sort('ticker').reset_index(drop=True)
pfTickers = pfTickers.merge(industry_data[['ticker', 'sector']])
sector_list = pfTickers['sector'].unique()
betas = Coef[['ticker', 'Mkt-RF']].merge(pfTickers).rename(columns={'Mkt-RF': 'beta'}).reset_index(drop=True)
mBeta = matrix(betas['beta'])
mCqaBeta = matrix(StockBeta.merge(pfTickers)['cqaBeta'])
longIndex = matrix(pfTickers.merge(longTickers, how='left').fillna(0)['bWeight'])
mLongIndex = np.diag(pfTickers.merge(longTickers, how='left').fillna(0)['bWeight'])
mLongIndex = matrix(mLongIndex[np.logical_or.reduce([np.sum(mLongIndex,1) > 0.5])]).trans()
# mLongIndex = matrix(np.diag(tickers.merge(longTickers, how='left').fillna(0)['bWeight']))
shortIndex = -matrix(pfTickers.merge(shortTickers, how='left').fillna(0)['bWeight'])
mShortIndex = -np.diag(pfTickers.merge(shortTickers, how='left').fillna(0)['bWeight'])
mShortIndex = matrix(mShortIndex[np.logical_or.reduce([np.sum(mShortIndex,1) > 0.5])]).trans()
# mShortIndex = matrix(np.diag(pfTickers.merge(shortTickers, how='left').fillna(0)['bWeight']))
sector_index = pfTickers[['ticker', 'sector']]
for sector in sector_list:
sector_index.loc[:,sector] = 0.0
sector_index.ix[sector_index['sector'] == sector, sector] = 1.0
mSector_index = matrix(sector_index.iloc[:, 2:].as_matrix())
# wTickers = functions.iniWeights(pfTickers, shortTickers, longTickers) # initial weights
wStart = matrix(functions.iniWeights(pfTickers, longTickers, shortTickers)['weight'])
N = pfTickers.shape[0]
id = spmatrix(1.0, range(N), range(N))
wBounds = matrix(np.ones((N,1)) * WEIGHT_BOUND)
longBounds = matrix(np.zeros((shortTickers.shape[0], 1)))
# longBounds = matrix(np.ones((shortTickers.shape[0], 1)) * 0.002)
shortBounds = matrix(np.zeros((longTickers.shape[0], 1)))
# shortBounds = matrix(np.ones((longTickers.shape[0], 1)) * (-0.005))
A = matrix([[mSector_index], [-mSector_index],
[mCqaBeta], [-mCqaBeta],
[longIndex], [-longIndex],
[shortIndex], [-shortIndex],
[id], [-id],
[-mLongIndex], [mShortIndex]]).trans()
b = matrix([SECTOR_WEIGHT, SECTOR_WEIGHT, SECTOR_WEIGHT, SECTOR_WEIGHT,
SECTOR_WEIGHT, SECTOR_WEIGHT, SECTOR_WEIGHT, SECTOR_WEIGHT,
SECTOR_WEIGHT, SECTOR_WEIGHT, SECTOR_WEIGHT, SECTOR_WEIGHT,
SECTOR_WEIGHT, SECTOR_WEIGHT, SECTOR_WEIGHT, SECTOR_WEIGHT,
BETA_BOUND, BETA_BOUND,
1, -0.98,
-0.98, 1,
wBounds, wBounds,
longBounds, shortBounds])
sol = solvers.lp(-mBeta, A, b)
w_res = sol['x']
print 'cqaBeta = %.4f' % np.float64(w_res.trans() * mCqaBeta)[0,0]
print 'beta = %.4f' % np.float64(w_res.trans() * mBeta)[0,0]
wTickers = pfTickers
wTickers['weight'] = w_res
return wTickers
def test_lp(self):
from cvxopt import solvers, glpk
c,G,h,A,b = self._prob_data
sol1 = solvers.lp(c,G,h)
self.assertTrue(sol1['status']=='optimal')
sol2 = solvers.lp(c,G,h,A,b)
self.assertTrue(sol2['status']=='optimal')
sol3 = solvers.lp(c,G,h,solver='glpk')
self.assertTrue(sol3['status']=='optimal')
sol4 = solvers.lp(c,G,h,A,b,solver='glpk')
self.assertTrue(sol4['status']=='optimal')
sol5 = glpk.lp(c,G,h)
self.assertTrue(sol5[0]=='optimal')
sol6 = glpk.lp(c,G,h,A,b)
self.assertTrue(sol6[0]=='optimal')
sol7 = glpk.lp(c,G,h,None,None)
self.assertTrue(sol7[0]=='optimal')
def test():
A = matrix([ [-1.0, -1.0, 0.0, 1.0], [1.0, -1.0, -1.0, -2.0] ])
print A
b = matrix([ 1.0, -2.0, 0.0, 4.0 ])
print b
c = matrix([ 2.0, 1.0 ])
print c
sol=solvers.lp(c,A,b)
#===Append Inverse Weight Vector===# C = numpy.c_[C,-1.0/Weights]; mC = numpy.c_[mC, -1.0/Weights]; #===Form Block Matrix===# Block = matrix(numpy.r_[C, mC]); #===Form RHS Vector===# d = matrix(numpy.asmatrix(numpy.r_[Desired, -Desired]).T); #===Form Cost Vector===# cost = matrix(numpy.r_[numpy.zeros(M+1), 1]); #===Solve LP===# sol=solvers.lp(cost,Block,d) #===Parse Solution===# a = sol['x'][0:-1]; delta = sol['x'][-1]; print "Delta: ", delta #===Make Filter===# design = list(a[-1:0:-1]); design.append(2.0*a[0]); temp = list(a[1::]); for t in temp: design.append(t); print "Filter Length: ", len(design), filter_length; #===Ensure Unity Gain===#
I[2 * ind] = ind
J[2 * ind] = listing[0]
val[2 * ind] = -1
I[2 * ind + 1] = ind
J[2 * ind + 1] = S + listing[1]
ctr = listing[2]
cvr = listing[3]
price = listing[4]
mcpc = listing[5]
val[2 * ind + 1] = -ctr * mcpc
b[ind] = -ctr * (mcpc + 0.03 * cvr * price)
ind += 1
I_2 = [total + i for i in range(S)]
J_2 = [i for i in range(S)]
val_2 = [-1 for i in range(S)]
b_2 = [0 for i in range(S)]
I_3 = [total + S + i for i in range(M)]
J_3 = [S + i for i in range(M)]
val_3 = [-1 for i in range(M)]
b_3 = [0 for i in range(M)]
mat = spmatrix(val + val_2 + val_3, I + I_2 + I_3, J + J_2 + J_3,
(total + S + M, S + M))
c = matrix([1 for i in range(S)] + budget_shop)
h = matrix(b + b_2 + b_3)
sol = solvers.lp(c, mat, h, feastol=1e-4)
np.save('/home/ubuntu/dual_sol/solution', sol['x'])
wgl = matrix(
np.loadtxt(file_group_weights, delimiter=",", skiprows=2, usecols=(1)))
# --------------------------------------------------------------------
# Find minimum cost food composition
# --------------------------------------------------------------------
# solve the linear program
# minimize: C'X
# subject to GX <= h, X >= 0
G = matrix([N, -N, -I, I])
h = scale * matrix([nh, -nl, -wl, wh])
# G = matrix([N,-N,-I,I,-group_def_matrix, group_def_matrix])
# h = matrix([nh,-nl,-wl,wh,-gwl,gwh])
X = solvers.lp(C, G, h)
# X=solvers.lp(C,G,h,solver='glpk')
# --------------------------------------------------------------------
# Find minimum weight food composition
# --------------------------------------------------------------------
# solve the linear probgam
# minimize: 1'*Y (i.e., total weight = y_1+y_2+ ...+y_{nf}, where 1 is a column vector of 1s, and 1' is its transpose)
# subject to: GY <= h, Y >= 0
D = matrix(np.ones((nf, 1)))
G = matrix([N, -N, -I, I, -group_def_matrix, group_def_matrix, C.trans()])
h = scale * matrix([nh, -nl, -wl, wh, -wgl, wgh, cost_hi])
Y = solvers.lp(D, G, h)
def solveLPWithDIC(ranking, k, dataSetName, algoName):
"""
Solve the linear program with DIC
@param ranking: list of candidate objects in the ranking
@param k: length of the ranking
@param dataSetName: Name of the data set the candidates are from
@param algoName: Name of inputed algorithm
return doubly stochastic matrix as numpy array
"""
print('Start building LP with DIC.')
#calculate the attention vector v using 1/log(1+indexOfRanking)
u = []
unproU = 0
proU = 0
proCount = 0
unproCount = 0
proListX = []
unproListX = []
for candidate in ranking[:k]:
u.append(candidate.learnedScores)
# initialize v with DCG
v = np.arange(1, (k + 1), 1)
v = 1 / np.log2(1 + v + 1)
v = np.reshape(v, (1, k))
arrayU = np.asarray(u)
#normalize input
arrayU = (arrayU - np.min(arrayU)) / (np.max(arrayU) - np.min(arrayU))
I = []
J = []
I2 = []
#set up indices for column and row constraints
for j in range(k**2):
J.append(j)
for i in range(k):
for j in range(k):
I.append(i)
for i in range(k):
for j in range(k):
I2.append(j)
for i in range(k):
if ranking[i].isProtected == True:
proCount += 1
proListX.append(i)
proU += arrayU[i]
else:
unproCount += 1
unproListX.append(i)
unproU += arrayU[i]
arrayU = np.reshape(arrayU, (k, 1))
uv = arrayU.dot(v)
uv = uv.flatten()
#negate objective function to convert maximization problem to minimization problem
uv = np.negative(uv)
# check if there are protected items
if proCount == 0:
print('Cannot create a P for ' + algoName + ' on data set ' +
dataSetName + ' because no protected items in data set.')
return 0, False
# check if there are unprotected items
if unproCount == 0:
print('Cannot create a P for ' + algoName + ' on data set ' +
dataSetName + ' because no unprotected items in data set.')
return 0, False
proU = proU / proCount
unproU = unproU / unproCount
initf = np.zeros((k, 1))
initf[proListX] = (1 / (proCount * proU)) * arrayU[proListX]
initf[unproListX] = (-(1 / (unproCount * unproU)) * arrayU[unproListX])
f1 = initf.dot(v)
f1 = f1.flatten()
f1 = np.reshape(f1, (1, k**2))
f = matrix(f1)
#set up constraints x <= 1
A = spmatrix(1.0, range(k**2), range(k**2))
#set up constraints x >= 0
A1 = spmatrix(-1.0, range(k**2), range(k**2))
#set up constraints that sum(rows)=1
M = spmatrix(1.0, I, J)
#set up constraints sum(columns)=1
M1 = spmatrix(1.0, I2, J)
#values for sums columns and rows == 1
h1 = matrix(1.0, (k, 1))
#values for x<=1
b = matrix(1.0, (k**2, 1))
#values for x >= 0
d = matrix(0.0, (k**2, 1))
#construct objective function
c = matrix(uv)
#assemble constraint matrix as sparse matrix
G = sparse([M, M1, A, A1, f])
#assemble constraint values
h = matrix([h1, h1, b, d, 0.0])
print('Start solving LP with DIC.')
try:
sol = solvers.lp(c, G, h)
except Exception:
print('Cannot create a P for ' + algoName + ' on data set ' +
dataSetName + '.')
return 0, False
print('Finished solving LP with DIC.')
return np.array(sol['x']), True
def solvep(self):
"""
Solve the Inverse RL problem
:return: Scalarization weights as array.
"""
n_states, n_actions, reward_dimension, gamma, P, R, pi, P_pi = self._prepare_variables(
)
v = self._prepare_v(n_states, n_actions, reward_dimension, P)
# weights for minimization term
c = np.vstack([
np.ones((n_states, 1)),
np.zeros((n_states * (n_actions - 1), 1)),
np.zeros((reward_dimension, 1))
])
# big block selection matrix
G = np.vstack([
np.hstack([
np.zeros((reward_dimension, n_states)),
np.zeros((reward_dimension, n_states * (n_actions - 1))),
np.eye(reward_dimension)
]),
np.hstack([
np.zeros((reward_dimension, n_states)),
np.zeros((reward_dimension, n_states * (n_actions - 1))),
-np.eye(reward_dimension)
]),
np.hstack([
np.vstack([
self._vertical_ones(n_actions - 1, n_states, i)
for i in xrange(n_states)
]),
np.eye(n_states * (n_actions - 1), n_states * (n_actions - 1)),
np.zeros((n_states * (n_actions - 1), reward_dimension))
]),
np.hstack([
np.zeros((n_states * (n_actions - 1), n_states)),
np.eye(n_states * (n_actions - 1), n_states * (n_actions - 1)),
np.vstack([
np.vstack([
-v[i, j, :].reshape(1, -1)
for j in xrange(n_actions - 1)
]) for i in xrange(n_states)
])
]),
np.hstack([
np.zeros((n_states * (n_actions - 1), n_states)), 2 *
np.eye(n_states * (n_actions - 1), n_states * (n_actions - 1)),
np.vstack([
np.vstack([
-v[i, j, :].reshape(1, -1)
for j in xrange(n_actions - 1)
]) for i in xrange(n_states)
])
])
])
# right hand side of inequalities
#h = np.vstack([np.ones((2 * reward_dimension, 1)), np.zeros(((n_states * (n_actions - 1)) * 3, 1))])
h = np.vstack([
np.ones((reward_dimension, 1)),
np.zeros((reward_dimension, 1)),
np.zeros(((n_states * (n_actions - 1)) * 3, 1))
])
solution = solvers.lp(matrix(-c), matrix(G), matrix(h))
alpha = np.asarray(solution['x'][-reward_dimension:])
return alpha.ravel()
#!/opt/local/bin/python2.7
from cvxopt import matrix, solvers
# Constraints
a = matrix([[1., 3., 3., -1., 0, 0], [-1., 2., 2., 0, -1., 0],
[1., 4., 0., 0., 0., -1.]])
b = matrix([20., 42., 30., 0., 0., 0.])
# Components of the functional
c = matrix([-5., -4., -6.])
# Solve the linear program
sol = solvers.lp(c, a, b)
print(sol['x'])
def path_decompose(a,
b,
a_true,
b_true,
overwrite_norm,
P,
use_GLPK,
sparsity=False):
'''This function takes in a node's information in the attempt to decompose it into the lowest number of paths that
accounts for the flow constraints.
Thhe algorithm uses many trials of a randomizaed optimization model and takes the best result seen.
**Do not use over_write_norm becuase a_true and b_true surrently have normalization instead of copycount.
a: a vector of the copytcounts of the in-edges for the node.
b: a vector of the copycounts of the out-edges for the node.
a_true: a vector that should have the copycount values for the in-edges but does not. (Don't Use)
b_true: a vector that should have the copycount values for the out-edges but does not. (Don't Use)
decides whether or not to use a_true and b_true.
This is a matrix of in-edges versus out-edges that has a 0 if there is a known path and a 1 otherwise.
'''
#mb_check = 1 #if this parameter is set to 1, if the data can be set using MB, then it
from cvxopt import matrix, solvers, spmatrix
import numpy, copy
from numpy import linalg as LA
from numpy import array
from numpy import zeros
import operator
solvers.options['msg_lev'] = 'GLP_MSG_OFF'
m = len(a) ## a is a vector of the current in edge copycount values.
n = len(b) ## b is a vector of the current out edge copycount values.
## Trivial case
if m == 0 or n == 0:
return [[], 0]
## Make all in flow values non-zero.
for i in range(m):
a[i] = max(a[i], 1e-10)
for j in range(n):
b[j] = max(b[j], 1e-10)
## If the flow in does not equal the flow out, make them equal.
if sum(a) > sum(b):
const = (sum(a) - sum(b)) / n
b = [k + const for k in b]
else:
const = (sum(b) - sum(a)) / m
a = [k + const for k in a]
## Trivial cases.
if m == 1:
answer = array(matrix(b, (m, n)))
return [answer, 0]
elif n == 1:
answer = array(matrix(a, (m, n)))
return [answer, 0]
A = matrix(0.,
(m + n - 1, m *
n)) ## A is used to enforce flow constraint on decomposition.
p = matrix(
0., (m * n, 1)
) ## This vector tells whether there is a known path for the pair of nodes or not.
for i in range(m):
for j in range(n):
A[i, j * m + i] = 1.
#check indexing
p[j * m + i] = P[i, j]
for j in range(n - 1): #Must be range(n) to get full mattrix
for i in range(m):
A[m + j, j * m + i] = 1.
z = a + b
rhs = matrix(z)
rhs = rhs[0:n + m - 1,
0] ## this vector is used to enforce flow constraints
weight = LA.norm(a, 1) ## (Not used)
eps = 0.001
tol = eps * weight #test for significance. Used for various purposes
sparsity_factor = 0.4 #very aggressive curently - revert to 0.1 later
removal_factor = 0.4
scale = max(max(rhs), 1e-100) * 0.01
#print(rhs)
trials = int(round(min(2 * m * n * max(m, n),
100))) ## Number of randomized trials.
curr_min = m * n + 1 ## The lowest number of non known paths seen so far.
curr_ans = []
## The best solution seen so far.
curr_err = 0
curr_mult = 0
## multiplicity of current solution
curr_on_unknown = 0
## amount of flow on non "known paths".
for ctr in range(
trials
): ## randomize the the coefficients for the known non=known path flow values.
c = matrix(abs(numpy.random.normal(0, 1, (m * n, 1))))
for i in range(m * n):
c[i] = c[i] * p[i]
## G and h are used to make sure that the flow values are non-negative.
G = spmatrix(-1., range(m * n), range(m * n))
h = matrix(0., (m * n, 1))
if use_GLPK:
sol = solvers.lp(c=c, G=G, h=h, A=A, b=rhs / scale, solver='glpk')
else:
sol = solvers.lp(c=c, G=G, h=h, A=A, b=rhs / scale)
temp_sol = array(sol['x']) * scale
another_sol = copy.deepcopy(temp_sol)
if overwrite_norm: ## Do not use right now because a_true and b_true are wrong.
#the true values of copy count are used to decide thresholding behavior
a = a_true[:]
b = b_true[:]
## This loop basically sets the flow values equal to 0 under certain conditions.
for i in range(m):
for j in range(n):
if another_sol[j * m + i] < sparsity_factor * min(a[i], b[j]):
another_sol[j * m + i] = 0
if temp_sol[j * m +
i] < removal_factor * min(a[i], b[j]) or temp_sol[
j * m + i] < tol: # temporarily disabled
temp_sol[j * m + i] = 0
another_sol[j * m + i] = 0
if another_sol[j * m + i] < 0:
another_sol[j * m + i] = 0
temp_sol[j * m + i] = 0
s = 0 ## s equals how many non-zero flows we are sending down non "known paths" in the temp solution
for i in range(m * n):
if p[i] > 0: #Only couont the paths that are not supported
s = s + numpy.count_nonzero(another_sol[i])
## if the temporary solution is less than the current minimum solution, replace current minimum solution with
## the temporary solution.
if s < curr_min:
curr_min = s ## current minimum value of non "known paths" in solution
curr_ans = temp_sol[:] ## answer that attains it.
curr_mult = 0 ## This is how many times a solution with this many non-zero non-known paths is seen.
curr_on_unknown = local_dot(
array(p), temp_sol
) ## This says how much flow we are sending down non "known paths"
else:
if s == curr_min:
if LA.norm(
curr_ans - temp_sol
) > tol: ## Determines whethee to classify the solutions as different.
curr_mult = curr_mult + 1
if curr_ans == [] or (
abs(sum(sum(temp_sol)) - sum(sum(curr_ans))) < tol and
(local_dot(array(p), temp_sol) < curr_on_unknown)) or sum(
sum(temp_sol)) > sum(sum(curr_ans)):
## These are a few conditions that make it the temporary solution:
## 1: curr_sol is empty. 2: total flow difference is below a threshold AND less flow is going down unknown paths. 3: total flow is greater than curr_ans total flow.
curr_ans = temp_sol[:]
curr_on_unknown = local_dot(array(p), temp_sol)
answer = matrix(0., (m, n))
if len(curr_ans) < m * n:
pdb.set_trace()
for i in range(m):
for j in range(n):
answer[i, j] = float(curr_ans[j * m + i])
# Uniqueness consideration
answer = array(answer)
non_unique = 0
if curr_mult > 1:
non_unique = 1
# This makes the flow solution more sparse
if sparsity != False:
if m * n > sparsity:
tmp_dict = {}
for i in range(m):
for j in range(n):
tmp_dict[(i, j)] = answer[i, j]
sorted_tmp = sorted(tmp_dict.items(),
key=operator.itemgetter(1))[::-1]
sorted_tmp = sorted_tmp[:sparsity]
new_ans = zeros((m, n))
for ind in sorted_tmp:
new_ans[ind[0][0], ind[0][1]] = ind[1]
answer = new_ans
return [answer, non_unique]
h[:, 3] = 1
# acceleration lower and upper
h[:, 4] = 20
h[:, 5] = 20
# Cost = 1/2 xPx + qx
cost = np.zeros((T, N))
cost[:, 2] = 1.0 # Theta cost
P = np.diag(cost.flatten())
#print(P)
# or maybe we should only care about the end
q = np.zeros((T, N))
q[:, 2] = -np.pi # offset
P = sparse(matrix(P))
q = matrix(q.flatten())
A = sparse(matrix(A.reshape((T * (N - 1), T * N))))
b = matrix(b.flatten())
G = sparse(matrix(G.reshape((T * 6, N * T))))
h = matrix(h.flatten())
# If sin(theta) is cost,
# cos(theta) would be a c vector for cost
# maybe an initial guess for x could help
# It all kind of seems too slow, but maybe glpk could help
for i in range(10):
lp(q, G, h, A, b)
#qp(P, q, G, h, A, b)
def outlierScore(neighbors, W, p, edges, C, s):
from cvxopt import matrix, solvers
""" Returns outlier score"""
pairs = list(itertools.combinations(neighbors, 2))
Hm = []
Lm = []
w = []
zeta = []
b = []
c = []
for i in range(p):
w.append([])
for rg in range(len(pairs)):
zeta.append([])
j = 0
for p in pairs: # Equation 3.9-3.12
n1 = min(p)
n2 = max(p)
if n2 in edges[
n1]: # Equation 3.9 and 3.10 (alternative lines for each)
Hm.append(-1.0)
Hm.append(0.0)
Lm.append(0.0)
Lm.append(0.0)
for i in range(len(w)):
w[i].append(1.0 * abs(W[n1][i] - W[n2][i]))
w[i].append(0.0)
for i in range(len(zeta)):
if (i == j):
zeta[i].append(-1.0)
zeta[i].append(-1.0)
else:
zeta[i].append(0.0)
zeta[i].append(0.0)
else: # Equation 3.11 and 3.12 (alternative lines for each)
Hm.append(0.0)
Hm.append(0.0)
Lm.append(1.0)
Lm.append(0.0)
for i in range(len(w)):
w[i].append(-1.0 * abs(W[n1][i] - W[n2][i]))
w[i].append(0.0)
for i in range(len(zeta)):
if (i == j):
zeta[i].append(-1.0)
zeta[i].append(-1.0)
else:
zeta[i].append(0.0)
zeta[i].append(0.0)
b.append(0.0)
b.append(0.0)
j += 1
for i in range(len(w)): # Equation 3.13
Hm.append(0.0)
Hm.append(0.0)
Lm.append(0.0)
Lm.append(0.0)
for i in range(len(zeta)):
zeta[i].append(0.0)
zeta[i].append(0.0)
for j in range(len(w)):
if (i == j):
w[j].append(1.0)
w[j].append(-1.0)
else:
w[j].append(0.0)
w[j].append(0.0)
b.append(1.0)
b.append(0.0)
# Equation 3.14
Hm.append(0.0)
Hm.append(0.0)
Lm.append(0.0)
Lm.append(0.0)
for i in range(len(zeta)):
zeta[i].append(0.0)
zeta[i].append(0.0)
for i in range(len(w)):
w[i].append(1.0)
w[i].append(-1.0)
b.append(1.0)
b.append(-1.0)
# Now populating c, the order of coefficients in c is Hm, Lm, w, zeta
c.append(1.0)
c.append(-1.0)
for i in w:
c.append(0.0)
for i in zeta:
c.append(1.0 * C / len(pairs))
tempList = []
tempList.append(Hm)
tempList.append(Lm)
for i in w:
tempList.append(i)
for i in zeta:
tempList.append(i)
A = matrix(tempList)
b = matrix(b)
c = matrix(c)
# Now feeding the input to simplex algo
# print A
# print b
# print c
try:
sol = solvers.lp(c, A, b, solver='glpk')
# print sol['x']
except Exception as e:
print("error")
print(e)
time.sleep(1)
return 0, None
if sol['x'] == None:
return None, None
else:
w1 = []
for i in range(len(w)):
w1.append([i, sol['x'][i + 2]])
w1 = sorted(w1, key=lambda x: x[1])
#print("w1:", w1)
return sol['x'][0] - sol['x'][1], [w1[i][0] for i in range(s)]
def ingredientsToRecipe(b, A, y):
sol = solvers.lp(b,A,y)
print('recipes', sol['x'])
def lp_irl(trans_probs, policy, gamma=0.5, l1=10, R_max=10):
"""
inputs:
trans_probs NxNxN_ACTIONS transition matrix
policy policy vector / map
R_max maximum possible value of recoverred rewards
gamma RL discount factor
l1 l1 regularization lambda
returns:
rewards Nx1 reward vector
"""
print np.shape(trans_probs)
N_STATES, _, N_ACTIONS = np.shape(trans_probs)
N_STATES = int(N_STATES)
N_ACTIONS = int(N_ACTIONS)
# Formulate a linear IRL problem
A = np.zeros([2 * N_STATES * (N_ACTIONS + 1), 3 * N_STATES])
b = np.zeros([2 * N_STATES * (N_ACTIONS + 1)])
c = np.zeros([3 * N_STATES])
for i in range(N_STATES):
a_opt = int(policy[i])
tmp_inv = np.linalg.inv(
np.identity(N_STATES) - gamma * trans_probs[:, :, a_opt])
cnt = 0
for a in range(N_ACTIONS):
if a != a_opt:
A[i * (N_ACTIONS - 1) + cnt, :N_STATES] = - \
np.dot(trans_probs[i, :, a_opt] - trans_probs[i, :, a], tmp_inv)
A[N_STATES * (N_ACTIONS - 1) + i * (N_ACTIONS - 1) + cnt, :N_STATES] = - \
np.dot(trans_probs[i, :, a_opt] - trans_probs[i, :, a], tmp_inv)
A[N_STATES * (N_ACTIONS - 1) + i * (N_ACTIONS - 1) + cnt,
N_STATES + i] = 1
cnt += 1
for i in range(N_STATES):
A[2 * N_STATES * (N_ACTIONS - 1) + i, i] = 1
b[2 * N_STATES * (N_ACTIONS - 1) + i] = R_max
for i in range(N_STATES):
A[2 * N_STATES * (N_ACTIONS - 1) + N_STATES + i, i] = -1
b[2 * N_STATES * (N_ACTIONS - 1) + N_STATES + i] = 0
for i in range(N_STATES):
A[2 * N_STATES * (N_ACTIONS - 1) + 2 * N_STATES + i, i] = 1
A[2 * N_STATES * (N_ACTIONS - 1) + 2 * N_STATES + i,
2 * N_STATES + i] = -1
for i in range(N_STATES):
A[2 * N_STATES * (N_ACTIONS - 1) + 3 * N_STATES + i, i] = 1
A[2 * N_STATES * (N_ACTIONS - 1) + 3 * N_STATES + i,
2 * N_STATES + i] = -1
for i in range(N_STATES):
c[N_STATES:2 * N_STATES] = -1
c[2 * N_STATES:] = l1
sol = solvers.lp(matrix(c), matrix(A), matrix(b))
rewards = sol['x'][:N_STATES]
rewards = normalize(rewards) * R_max
return rewards
def optimize(sr_grid,
expense,
W,
init=None,
eps=1e-2,
verbose=False,
max_iter=10000):
solvers.options['show_progress'] = False
solvers.options['glpk'] = dict(msg_lev='GLP_MSG_OFF')
n = sr_grid.shape[0]
d = np.zeros(n + 1)
d[0] = 1.
# A1 ... sum(p) = 1
A1 = np.ones((1, n + 1))
A1[0, 0] = 0.
A1 = A1
b = np.array([1.])
stimulus_pdfs = []
if init is None:
new_pdf = np.ones(n)
new_pdf = new_pdf / new_pdf.sum()
elif all(init > 0) and np.isclose(np.sum(init), 1.):
if len(init) == n:
new_pdf = init
else:
raise ValueError(
'init has incorrect size. {} expected but len(init) is {}'.
format(n, len(init)))
else:
raise ValueError('init is not a probability distribution')
stimulus_pdfs.append(new_pdf)
sensitivity_matrix = np.zeros((max_iter, n + 1))
sensitivity_matrix[:, 0] = 1.
# G0 ... p > 0
G0 = -np.eye(sr_grid.shape[0], sr_grid.shape[0] + 1, 1, dtype=float)
cost_matrix = np.concatenate(([[0]], [expense]), axis=1)
G = np.concatenate((cost_matrix, G0, sensitivity_matrix))
# G = np.concatenate((G0, sensitivity_matrix))
h = np.zeros(G.shape[0], dtype=float)
h[0] = W
min_capacity = 0
for i in range(max_iter):
prior = sr_grid.T.dot(stimulus_pdfs[i])
new_sensitivity = sensitivity(sr_grid, prior)
lim = 1e2
new_sensitivity[new_sensitivity > lim] = lim
sensitivity_matrix[i, 1:] = -new_sensitivity
# G[G0.shape[0] + i, 1:] = -new_sensitivity
G[G0.shape[0] + i + 1, 1:] = -new_sensitivity
# row_count = G0.shape[0] + i + 1
row_count = G0.shape[0] + i + 2
if i >= 0:
res = solvers.lp(matrix(-d),
matrix(G[:row_count]),
matrix(h[:row_count]),
matrix(A1),
matrix(b),
solver=None)
else:
res = solvers.lp(-d,
G[:row_count],
h[:row_count],
A1,
b,
solver=None)
# print(res)
stimulus_pdfs.append(np.array(res['x']).T[0, 1:])
prior = sr_grid.T.dot(stimulus_pdfs[-1])
new_sensitivity = sensitivity(sr_grid, prior)
lim = 1e2
new_sensitivity[new_sensitivity > lim] = lim
min_capacity = (new_sensitivity * stimulus_pdfs[-1]).sum()
max_capacity = -res['primal objective']
# max_capacity = stimulus
capacity = min_capacity
if min_capacity > 0:
accuracy = (max_capacity - min_capacity) / capacity
else:
accuracy = np.inf
if verbose is True:
print('min: {}, max: {}, accuracy: {}'.format(
min_capacity, max_capacity,
(max_capacity - min_capacity) / min_capacity))
if accuracy < eps:
break
else:
warnings.warn(f'Maximum number of iterations ({max_iter}) reached',
RuntimeWarning)
capacity = (min_capacity + max_capacity) / 2
out_pdf = sr_grid.T.dot(stimulus_pdfs[-1])
return {
'pdf': stimulus_pdfs[-1],
'fun': capacity,
'sensitivity': sensitivity(sr_grid, out_pdf)
}
def llp_irl(sf, M, k, T, phi, *, N=1000, p=2.0, verbose=False):
"""
Implements Linear Programming IRL for large state spaces by NG and
Russell, 2000
See https://thinkingwires.com/posts/2018-02-13-irl-tutorial-1.html for a
good reference.
@param sf - A state factory function that takes no arguments and returns
an i.i.d. sample from the MDP state space
@param M - The number of sub-samples to draw from the state space |S_0|
when estimating the expert's reward function
@param k - The number of actions |A|
@param T - A sampling transition function T(s, ai) -> s' encoding a
stationary deterministic policy. The structure of T must be that the
0th action T(:, 0) corresponds to a sample from the expert policy, and
T(:, i), i != 0 corresponds to a sample from the ith non-expert action
at each state, for some arbitrary but consistent ordering of states
@param phi - A vector of d basis functions phi_i(s) mapping from S to real
numbers
@param N - Number of transition samples to use when computing expectations
over the Value basis functions
@param p - Penalty function coefficient. Ng and Russell find p=2 is robust
Must be >= 1
@param verbose - If true, progress information will be shown
@return A vector of d 'alpha' coefficients for the basis functions phi(S)
that allows rewards to be computed for a state via the inner product
alpha_i · phi
@return A result object from the optimiser
"""
# Measure number of basis functions
d = len(phi)
# Enforce valid penalty function coefficient
assert p >= 1, \
"Penalty function coefficient must be >= 1, was {}".format(p)
def compute_value_expectation_tensor(sf, M, k, T, phi, N, verbose):
"""
Computes the value expectation tensor VE
This is an array of shape (d, k-1, M) where VE[:, i, j] is a vector of
coefficients that, when multiplied with the alpha vector give the
expected difference in value between the expert policy action and the
ith action from state s_j
@param sf - A state factory function that takes no arguments and returns
an i.i.d. sample from the MDP state space
@param M - The number of sub-samples to draw from the state space |S_0|
when estimating the expert's reward function
@param k - The number of actions |A|
@param T - A sampling transition function T(s, ai) -> s' encoding a
stationary deterministic policy. The structure of T must be that the
0th action T(:, 0) corresponds to a sample from the expert policy, and
T(:, i), i != 0 corresponds to a sample from the ith non-expert action
at each state, for some arbitrary but consistent ordering of states
@param phi - A vector of d basis functions phi_i(s) mapping from S to real
numbers
@param N - Number of transition samples to use when computing expectations
over the value basis functions
@param verbose - If true, progress information will be shown
@return The value expectation tensor VE. A numpy array of shape
(d, k-1, M)
"""
def expectation(fn, sf, N):
"""
Helper function to estimate an expectation over some function fn(sf())
@param fn - A function of a single variable that the expectation will
be computed over
@param sf - A state factory function - takes no variables and returns
an i.i.d. sample from the state space
@param N - The number of draws to use when estimating the expectation
@return An estimate of the expectation E[fn(sf())]
"""
state = sf()
return sum([fn(sf()) for n in range(N)]) / N
# Measure number of basis functions
d = len(phi)
# Prepare tensor
VE_tensor = np.zeros(shape=(d, k-1, M))
# Draw M initial states from the state space
for j in range(M):
if verbose: print("{} / {}".format(j, M))
s_j = sf()
# Compute E[phi(s')] where s' is drawn from the expert policy
expert_basis_expectations = np.array([
expectation(phi[di], lambda: T(s_j, 0), N) for di in range(d)
])
# Loop over k-1 non-expert actions
for i in range(1, k):
# Compute E[phi(s')] where s' is drawn from the ith non-expert action
ith_non_expert_basis_expectations = np.array([
expectation(phi[di], lambda: T(s_j, i), N) for di in range(d)
])
# Compute and store the expectation difference for this initial
# state
VE_tensor[:, i-1, j] = expert_basis_expectations - \
ith_non_expert_basis_expectations
return VE_tensor
# Precompute the value expectation tensor VE
# This is an array of shape (d, k-1, M) where VE[:, i, j] is a vector of
# coefficients that, when multiplied with the alpha vector give the
# expected difference in value between the expert policy action and the
# ith action from state s_j
if verbose: print("Computing expectations...")
VE_tensor = compute_value_expectation_tensor(sf, M, k, T, phi, N, verbose)
# Formulate the linear programming problem constraints
# NB: The general form for adding a constraint looks like this
# c, A_ub, b_ub = f(c, A_ub, b_ub)
if verbose: print("Composing LP problem...")
def add_costly_single_step_constraints(c, A_ub, b_ub):
"""
Augments the objective and adds constraints to implement the Linear
Programming IRL method for large state spaces
This will add M extra variables and 2M*(k-1) constraints
NB: Assumes the true optimisation variables are first in the c vector
"""
# Step 1: Add the extra optimisation variables for each min{} operator
# (one per sampled state)
c = np.hstack([np.zeros(shape=(1, d)), np.ones(shape=(1, M))])
A_ub = np.hstack([A_ub, np.zeros(shape=(A_ub.shape[0], M))])
# Step 2: Add the constraints
# Loop for each of the starting sampled states s_j
for j in range(M):
if verbose: print("CSS: Adding constraints ({:.2f}%)".format(j/M*100))
# Loop over the k-1 non-expert actions
for i in range(1, k):
# Add two constraints, one for each half of the penalty
# function p(x)
constraint_row = np.hstack([VE_tensor[:, i-1, j], \
np.zeros(shape=M)])
constraint_row[d + j] = -1
A_ub = np.vstack((A_ub, constraint_row))
b_ub = np.vstack((b_ub, 0))
constraint_row = np.hstack([p * VE_tensor[:, i-1, j], \
np.zeros(shape=M)])
constraint_row[d + j] = -1
A_ub = np.vstack((A_ub, constraint_row))
b_ub = np.vstack((b_ub, 0))
return c, A_ub, b_ub
def add_alpha_size_constraints(c, A_ub, b_ub):
"""
Add constraints for a maximum |alpha| value of 1
This will add 2 * d extra constraints
NB: Assumes the true optimisation variables are first in the c vector
"""
for i in range(d):
constraint_row = [0] * A_ub.shape[1]
constraint_row[i] = 1
A_ub = np.vstack((A_ub, constraint_row))
b_ub = np.vstack((b_ub, 1))
constraint_row = [0] * A_ub.shape[1]
constraint_row[i] = -1
A_ub = np.vstack((A_ub, constraint_row))
b_ub = np.vstack((b_ub, 1))
return c, A_ub, b_ub
# Prepare LP constraint matrices
c = np.zeros(shape=[1, d], dtype=float)
A_ub = np.zeros(shape=[0, d], dtype=float)
b_ub = np.zeros(shape=[0, 1])
# Compose LP optimisation problem
c, A_ub, b_ub = add_costly_single_step_constraints(c, A_ub, b_ub)
c, A_ub, b_ub = add_alpha_size_constraints(c, A_ub, b_ub)
if verbose:
print("Number of optimisation variables: {}".format(c.shape[1]))
print("Number of constraints: {}".format(A_ub.shape[0]))
# Solve for a solution
if verbose: print("Solving LP problem...")
# NB: cvxopt.solvers.lp expects a 1d c vector
solvers.options['show_progress'] = verbose
res = solvers.lp(matrix(c[0, :]), matrix(A_ub), matrix(b_ub))
# Extract the true optimisation variables
alpha_vector = res['x'][0:d].T
return alpha_vector, res
def linear_program(c, G, h, A=None, b=None, solver=None):
"""
Solves the dual linear programs:
- Minimize `c'x` subject to `Gx + s = h`, `Ax = b`, and `s \geq 0` where
`'` denotes transpose.
- Maximize `-h'z - b'y` subject to `G'z + A'y + c = 0` and `z \geq 0`.
INPUT:
- ``c`` -- a vector
- ``G`` -- a matrix
- ``h`` -- a vector
- ``A`` -- a matrix
- ``b`` --- a vector
- ``solver`` (optional) --- solver to use. If None, the cvxopt's lp-solver
is used. If it is 'glpk', then glpk's solver
is used.
These can be over any field that can be turned into a floating point
number.
OUTPUT:
A dictionary ``sol`` with keys ``x``, ``s``, ``y``, ``z`` corresponding
to the variables above:
- ``sol['x']`` -- the solution to the linear program
- ``sol['s']`` -- the slack variables for the solution
- ``sol['z']``, ``sol['y']`` -- solutions to the dual program
EXAMPLES:
First, we minimize `-4x_1 - 5x_2` subject to `2x_1 + x_2 \leq 3`,
`x_1 + 2x_2 \leq 3`, `x_1 \geq 0`, and `x_2 \geq 0`::
sage: c=vector(RDF,[-4,-5])
sage: G=matrix(RDF,[[2,1],[1,2],[-1,0],[0,-1]])
sage: h=vector(RDF,[3,3,0,0])
sage: sol=linear_program(c,G,h)
sage: sol['x']
(0.999..., 1.000...)
Next, we maximize `x+y-50` subject to `50x + 24y \leq 2400`,
`30x + 33y \leq 2100`, `x \geq 45`, and `y \geq 5`::
sage: v=vector([-1.0,-1.0,-1.0])
sage: m=matrix([[50.0,24.0,0.0],[30.0,33.0,0.0],[-1.0,0.0,0.0],[0.0,-1.0,0.0],[0.0,0.0,1.0],[0.0,0.0,-1.0]])
sage: h=vector([2400.0,2100.0,-45.0,-5.0,1.0,-1.0])
sage: sol=linear_program(v,m,h)
sage: sol['x']
(45.000000..., 6.2499999...3, 1.00000000...)
sage: sol=linear_program(v,m,h,solver='glpk')
sage: sol['x']
(45.0..., 6.25, 1.0...)
"""
from cvxopt.base import matrix as m
from cvxopt import solvers
solvers.options['show_progress'] = False
if solver == 'glpk':
from cvxopt import glpk
glpk.options['LPX_K_MSGLEV'] = 0
c_ = m(c.base_extend(RDF).numpy())
G_ = m(G.base_extend(RDF).numpy())
h_ = m(h.base_extend(RDF).numpy())
if A != None and b != None:
A_ = m(A.base_extend(RDF).numpy())
b_ = m(b.base_extend(RDF).numpy())
sol = solvers.lp(c_, G_, h_, A_, b_, solver=solver)
else:
sol = solvers.lp(c_, G_, h_, solver=solver)
status = sol['status']
if status != 'optimal':
return {
'primal objective': None,
'x': None,
's': None,
'y': None,
'z': None,
'status': status
}
x = vector(RDF, list(sol['x']))
s = vector(RDF, list(sol['s']))
y = vector(RDF, list(sol['y']))
z = vector(RDF, list(sol['z']))
return {
'primal objective': sol['primal objective'],
'x': x,
's': s,
'y': y,
'z': z,
'status': status
}
def cc_lp(W):
print('Inside the cc_lp function of cc.py^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n')
m = W.shape[0]
W.shape = m*m,1
#reflexivity constraints
G1 = zeros((m,m*m))
r = array([i*m+i for i in arange(m)])
G1[arange(m),r] = 1
h1 = ones(m)
G11 = zeros((m,m*m))
G11[arange(m),r] = -1
h11 = -ones(m)
ids=array([[i,j] for i in arange(m) for j in arange(m)])
rc=ids[where(ids[:,0]<ids[:,1])]
r=array([i[0]*m+i[1] for i in rc])
# symmetry constraints
nc = len(rc)
G2 = zeros((nc,m*m))
G2[arange(nc),r] = 1
rt = array([i[1]*m+i[0] for i in rc])
G2[arange(nc),rt] = -1
h2 = zeros(nc)
G21 = zeros((nc,m*m))
G21[arange(nc),r] = -1
G21[arange(nc),rt] = 1
h21 = zeros(nc)
#range [0,1] constraints
nc = len(r)
G4 = zeros((nc,m*m))
G4[arange(nc),r] = -1
h4 = zeros(nc)
G41 = zeros((nc,m*m))
G41[arange(nc),r] = 1
h41 = ones(nc)
# transitivity constraints
ids = array([[i,j,k] for i in arange(m) for j in arange(m) for k in arange(m)])
ids = ids[where(ids[:,0] != ids[:,1])]
ids = ids[where(ids[:,1] != ids[:,2])] # there has to be a better way to do this...
rc = ids[where(ids[:,2] != ids[:,0])]
nc = len(rc)
r1=array([i[0]*m+i[1] for i in rc])
r2=array([i[1]*m+i[2] for i in rc])
r3=array([i[2]*m+i[0] for i in rc])
G3 = zeros((nc,m*m))
G3[arange(nc),r1] = 1
G3[arange(nc),r2] = 1
G3[arange(nc),r2] = -1
h3 = ones(nc)
G = vstack((G1,G11,G2,G21,G3,G4,G41))
h = hstack((h1,h11,h2,h21,h3,h4,h41))
sol = solvers.lp(matrix(-W), matrix(G), matrix(h))
sol = array(sol['x'])
sol.shape = m,m
return sol
def apply_constraints(operator, n_fermions, use_scipy=True):
"""Function to use linear programming to apply constraints.
Args:
operator(FermionOperator): FermionOperator with only 1- and 2-body
terms that we wish to vectorize.
n_fermions(int): The number of particles in the simulation.
use_scipy(bool): Whether to use scipy (True) or cvxopt (False).
Returns:
modified_operator(FermionOperator): The operator with reduced norm
that has been modified with equality constraints.
"""
# Get constraint matrix.
n_orbitals = count_qubits(operator)
constraints = constraint_matrix(n_orbitals, n_fermions)
n_constraints, n_terms = constraints.get_shape()
# Get vectorized operator.
vectorized_operator = operator_to_vector(operator)
initial_bound = numpy.sum(numpy.absolute(vectorized_operator[1::]))**2
print('Initial bound on measurements is %f.' % initial_bound)
# Get linear programming coefficient vector.
n_variables = n_constraints + n_terms
lp_vector = numpy.zeros(n_variables, float)
lp_vector[-n_terms:] = 1.
# Get linear programming constraint matrix.
lp_constraint_matrix = scipy.sparse.dok_matrix((2 * n_terms, n_variables))
for (i, j), value in constraints.items():
if j:
lp_constraint_matrix[j, i] = value
lp_constraint_matrix[n_terms + j, i] = -value
for i in range(n_terms):
lp_constraint_matrix[i, n_constraints + i] = -1.
lp_constraint_matrix[n_terms + i, n_constraints + i] = -1.
# Get linear programming constraint vector.
lp_constraint_vector = numpy.zeros(2 * n_terms, float)
lp_constraint_vector[:n_terms] = vectorized_operator
lp_constraint_vector[-n_terms:] = -vectorized_operator
# Perform linear programming.
print('Starting linear programming.')
if use_scipy:
options = {'maxiter': int(1e6)}
bound = n_constraints * [(None, None)] + n_terms * [(0, None)]
solution = scipy.optimize.linprog(c=lp_vector,
A_ub=lp_constraint_matrix.toarray(),
b_ub=lp_constraint_vector,
bounds=bound,
options=options)
# Analyze results.
print(solution['message'])
assert solution['success']
solution_vector = solution['x']
objective = solution['fun']**2
print('Program terminated after %i iterations.' % solution['nit'])
else:
# Convert to CVXOpt sparse matrix.
from cvxopt import matrix, solvers, spmatrix
lp_vector = matrix(lp_vector)
lp_constraint_matrix = lp_constraint_matrix.tocoo()
lp_constraint_matrix = spmatrix(lp_constraint_matrix.data,
lp_constraint_matrix.row.tolist(),
lp_constraint_matrix.col.tolist())
lp_constraint_vector = matrix(lp_constraint_vector)
# Run linear programming.
solution = solvers.lp(c=lp_vector,
G=lp_constraint_matrix,
h=lp_constraint_vector,
solver='glpk')
# Analyze results.
print(solution['status'])
solution_vector = numpy.array(solution['x']).transpose()[0]
# Alternative bound.
residuals = solution_vector[-n_terms:]
alternative_bound = numpy.sum(numpy.absolute(residuals[1::]))**2
print('Bound implied by solution vector is %f.' % alternative_bound)
# Make sure residuals are positive.
for residual in residuals:
assert residual > -1e-6
# Get bound on updated Hamiltonian.
weights = solution_vector[:n_constraints]
final_vectorized_operator = (vectorized_operator -
constraints.transpose() * weights)
final_bound = numpy.sum(numpy.absolute(final_vectorized_operator[1::]))**2
print('Actual bound determined is %f.' % final_bound)
# Return modified operator.
modified_operator = vector_to_operator(final_vectorized_operator,
n_orbitals)
return (modified_operator + hermitian_conjugated(modified_operator)) / 2.
def solvealge(self):
"""
Solve the Inverse RL problem
:return: Scalarization weights as array.
"""
n_states, n_actions, reward_dimension, gamma, P, R, pi, P_pi = self._prepare_variables(
)
v = self._prepare_v(n_states, n_actions, reward_dimension, P)
D = reward_dimension
x_size = n_states + (n_actions - 1) * n_states * 2 + D
c = -np.hstack([np.ones(n_states), np.zeros(x_size - n_states)])
assert c.shape[0] == x_size
A = np.vstack([
np.hstack([
np.zeros((n_states * (n_actions - 1), n_states)),
np.eye(n_states * (n_actions - 1)),
-np.eye(n_states * (n_actions - 1)),
np.vstack([
-v[i, j, :].reshape(1, -1) for i in range(n_states)
for j in range(n_actions - 1)
])
]),
# np.hstack([
# np.zeros((1, x_size - D)),
# np.ones((1, D))
# ])
])
assert A.shape[1] == x_size
b = np.vstack([
np.zeros((A.shape[0], 1)),
# np.zeros((A.shape[0] - 1, 1)),
# np.ones((1, 1))
])
# p-Function
n = 2.0
bottom_row = np.vstack([
np.hstack([
np.ones((n_actions - 1, 1)).dot(np.eye(1, n_states, l)),
np.hstack([
-np.eye(n_actions - 1) if i == l else np.zeros(
(n_actions - 1, n_actions - 1))
for i in range(n_states)
]),
np.hstack([
n * np.eye(n_actions - 1) if i == l else np.zeros(
(n_actions - 1, n_actions - 1))
for i in range(n_states)
]),
np.zeros((n_actions - 1, D))
]) for l in range(n_states)
])
assert bottom_row.shape[1] == x_size
G = np.vstack([
# np.hstack([
# np.zeros((1, n_states)),
# np.zeros((1, n_states * (n_actions - 1))),
# np.zeros((1, n_states * (n_actions - 1))),
# np.ones((1, D))]),
np.hstack([
np.zeros((D, n_states)),
np.zeros((D, n_states * (n_actions - 1))),
np.zeros((D, n_states * (n_actions - 1))),
np.eye(D)
]),
np.hstack([
np.zeros((D, n_states)),
np.zeros((D, n_states * (n_actions - 1))),
np.zeros((D, n_states * (n_actions - 1))), -np.eye(D)
]),
np.hstack([
np.zeros((n_states * (n_actions - 1), n_states)),
-np.eye(n_states * (n_actions - 1)),
np.zeros(
(n_states * (n_actions - 1), n_states * (n_actions - 1))),
np.zeros((n_states * (n_actions - 1), D))
]),
np.hstack([
np.zeros((n_states * (n_actions - 1), n_states)),
np.zeros(
(n_states * (n_actions - 1), n_states * (n_actions - 1))),
-np.eye(n_states * (n_actions - 1)),
np.zeros((n_states * (n_actions - 1), D))
]),
bottom_row
])
assert G.shape[1] == x_size
# h = np.vstack([np.ones((D * 2, 1)),
# np.zeros((n_states * (n_actions - 1) * 2 + bottom_row.shape[0], 1))])
# slack tuning variable
m = 1.0
# m = 10.0
# h = np.vstack([np.ones((1, 1)), m * np.ones((D, 1)), np.zeros((D, 1)),
# np.zeros((n_states * (n_actions - 1) * 2 + bottom_row.shape[0], 1))])
h = np.vstack([
m * np.ones((D, 1)),
np.ones((D, 1)),
np.zeros((n_states * (n_actions - 1) * 2 + bottom_row.shape[0], 1))
])
# c = c.reshape(-1, 1)
# b = b.reshape(-1, 1)
# print c.shape
# print G.shape
# print h.shape
# print A.shape
# print b.shape
# normalize each row
# asum = np.sum(np.abs(A), axis=1)
# print asum.shape
# print A.shape
# A /= asum[:, np.newaxis]
# b /= asum
# solvers.options['feastol'] = 1e-1
# solvers.options['abstol'] = 1e-3
# solvers.options['show_progress'] = True
# c = matrix(c)
# G = matrix(G)
# h = matrix(h)
# A = matrix(A)
# b = matrix(b)
solution = solvers.lp(matrix(c), matrix(G), matrix(h), matrix(A),
matrix(b))
alpha = np.asarray(solution['x'][-reward_dimension:], dtype=np.double)
return alpha.ravel()
finaldata.append(temp) A = np.array(finaldata) * (-1) n = len(finaldata) m = len(finaldata[0]) #根据之前所述构造矩阵 a1 = np.concatenate((A, (-1) * np.eye(n)), axis=1) a2 = np.concatenate((np.zeros([n, m]), (-1) * np.eye(n)), axis=1) A1 = np.concatenate((a1, a2)) c1 = np.array([0.0] * m + [1.0] * n) b1 = np.array([-1.0] * n + [0.0] * n) #带入算法求解 c1 = matrix(c1) A1 = matrix(A1) b1 = matrix(b1) sol1 = solvers.lp(c1, A1, b1) w2 = list(sol1['x']) x1 = [0, 1] y1 = [-(w2[0] + w2[1] * i) / w2[2] for i in x1] plt.scatter(X, Y) plt.plot(x, y, 'r') plt.show()
#程序文件Pex5_4.py
import numpy as np
from cvxopt import matrix, solvers
c=matrix([-4.,-5]); A=matrix([[2.,1],[1,2],[-1,0],[0,-1]]).T
b=matrix([3.,3,0,0]); sol=solvers.lp(c,A,b)
print("最优解为：\n",sol['x'])
print("最优值为：",sol['primal objective'])
q[index][:, index].shape
r[index].shape
newA = q[index][:, index].dot(r[index])
import scipy.sparse
sparseG = scipy.sparse.csc_matrix(G)
u, s, vt = scipy.sparse.linalg.svds(sparseG)
from cvxopt import sparse, spmatrix, matrix, solvers
sparse(matrix(G))
sol = solvers.lp(matrix(c), sparse(matrix(G)), matrix(h), sparse(matrix(A)),
matrix(b))
sol1 = solvers.lp(matrix(c), sparse(matrix(G)), matrix(h), sparse(matrix(A)),
matrix(b), 'glpk')
sol1 = solvers.lp(matrix(c), sparse(matrix(G)), matrix(h),
sparse(matrix(newA)), matrix(b), 'glpk')
print(sol['x'])
c = matrix([-4., -5.])
G = matrix([[2., 1., -1., 0.], [1., 2., 0., -1.]])
h = matrix([3., 3., 0., 0.])
sol = solvers.lp(c, G, h)
sol = solvers.lp(c, G, h, None, None, 'glpk')
print(sol['x'])
def cvar_regression(basis_matrix, values, alpha, verbosity=1):
from cvxopt import matrix, solvers, spmatrix
# do not include constant basis in optimization
assert alpha < 1 and alpha > 0
basis_matrix = basis_matrix[:, 1:]
assert basis_matrix.ndim == 2
assert values.ndim == 1
nsamples, nbasis = basis_matrix.shape
assert values.shape[0] == nsamples
active_index = int(np.ceil(alpha*nsamples)) - \
1 # 0 based index 0,...,nsamples-1
nactive_samples = nsamples - (active_index + 1)
assert nactive_samples > 0, ('no samples in alpha quantile')
beta = np.arange(1, nsamples + 1, dtype=float) / nsamples
beta[active_index - 1] = alpha
beta_diff = np.diff(beta[active_index - 1:-1])
# print (beta_diff
assert beta_diff.shape[0] == nactive_samples
v_coef = np.log(1 - beta[active_index -
1:-2]) - np.log(1 - beta[active_index:-1])
v_coef /= nsamples * (1 - alpha)
nvconstraints = nsamples * nactive_samples
Iv = np.identity(nvconstraints)
Isamp = np.identity(nsamples)
Iactsamp = np.identity(nactive_samples)
# nactive_samples = p
# nsamples = m
# nbasis = n
# design vars [c_1,...,c_n,u1,...,u_{m-p},v_1,...,v_{m-p}m,w]
c_arr = np.hstack((
basis_matrix.sum(axis=0) / nsamples,
1 / (1 - alpha) * beta_diff,
# np.tile(v_coef,nsamples), # tile([1,2],2) = [1,2,1,2]
np.repeat(v_coef, nsamples), # repeat([1,2],2) = [1,1,2,2]
1. / (nsamples * (1 - alpha)) * np.ones(1)))
num_opt_vars = nbasis + nactive_samples + nvconstraints + 1
# v_ij variables ordering: loop through j fastest, e.g. v_11,v_{12} etc
# v_ij+h'c+u_i <=y_j
constraints_1 = np.hstack((-np.tile(basis_matrix, (nactive_samples, 1)),
-np.repeat(Iactsamp, nsamples, axis=0), -Iv,
np.zeros((nvconstraints, 1))))
# W+h'c<=y_j
constraints_2 = np.hstack(
(-basis_matrix, np.zeros(
(nsamples, nactive_samples)), np.zeros(
(nsamples, nvconstraints)), -np.ones((nsamples, 1))))
# v_ij >=0
constraints_3 = np.hstack((np.zeros(
(nvconstraints, nbasis + nactive_samples)), -Iv,
np.zeros((nvconstraints, 1))))
# print ((constraints_1.shape, constraints_2.shape, constraints_3.shape)
G_arr = np.vstack((constraints_1, constraints_2, constraints_3))
h_arr = np.hstack(
(-np.tile(values, nactive_samples), -values, np.zeros(nvconstraints)))
assert G_arr.shape[1] == num_opt_vars
assert G_arr.shape[0] == h_arr.shape[0]
assert c_arr.shape[0] == num_opt_vars
c = matrix(c_arr)
#G = matrix(G_arr)
h = matrix(h_arr)
I, J, data = sparse.find(G_arr)
G = spmatrix(data, I, J, size=G_arr.shape)
if verbosity < 1:
solvers.options['show_progress'] = False
else:
solvers.options['show_progress'] = True
# solvers.options['abstol'] = 1e-10
# solvers.options['reltol'] = 1e-10
# solvers.options['feastol'] = 1e-10
sol = np.asarray(solvers.lp(c=c, G=G, h=h)['x'])[:nbasis]
residuals = values - basis_matrix.dot(sol)[:, 0]
coef = np.append(conditional_value_at_risk(residuals, alpha), sol)
return coef
import numpy as np
'''
problem:
mimimize 2 x1 + x2
subject to
-x1 + x2 <= 1
x1 + x2 >= 2
x2 >= 0
x1 - 2x2 <= 4
'''
# only can use <=, can't use >=, so change notion
c = matrix([2.0, 1.0])
b = matrix([1.0, -2.0, 0.0, 4.0])
A = matrix([[-1.0, -1.0, 0.0, 1.0], [1.0, -1.0, -1.0, -2.0]])
sol = solvers.lp(c, A, b)
print(sol)
print(sol['x'])
print(sol['primal objective'])
'''
run result:
pcost dcost gap pres dres k/t
0: 2.6471e+00 -7.0588e-01 2e+01 8e-01 2e+00 1e+00
1: 3.0726e+00 2.8437e+00 1e+00 1e-01 2e-01 3e-01
2: 2.4891e+00 2.4808e+00 1e-01 1e-02 2e-02 5e-02
3: 2.4999e+00 2.4998e+00 1e-03 1e-04 2e-04 5e-04
4: 2.5000e+00 2.5000e+00 1e-05 1e-06 2e-06 5e-06
5: 2.5000e+00 2.5000e+00 1e-07 1e-08 2e-08 5e-08
Optimal solution found.
{'primal infeasibility': 1.1368786420268754e-08,
def cvar_regression_quadrature(basis_matrix,
values,
alpha,
nquad_intervals,
verbosity=1,
trapezoid_rule=False,
solver_name='cvxopt'):
"""
solver_name = 'cvxopt'
solver_name='glpk'
trapezoid works but default option is better.
"""
assert alpha < 1 and alpha > 0
basis_matrix = basis_matrix[:, 1:]
assert basis_matrix.ndim == 2
assert values.ndim == 1
nsamples, nbasis = basis_matrix.shape
assert values.shape[0] == nsamples
if not trapezoid_rule:
# left-hand piecewise constant quadrature rule
beta = np.linspace(alpha, 1,
nquad_intervals + 2)[:-1] # quadrature points
dx = beta[1] - beta[0]
weights = dx * np.ones(beta.shape[0])
nuvars = weights.shape[0]
nvconstraints = nsamples * nuvars
num_opt_vars = nbasis + nuvars + nvconstraints
num_constraints = 2 * nsamples * nuvars
else:
beta = np.linspace(alpha, 1, nquad_intervals + 1) # quadrature points
dx = beta[1] - beta[0]
weights = dx * np.ones(beta.shape[0])
weights[0] /= 2
weights[-1] /= 2
weights = weights[:-1] # ignore left hand side
beta = beta[:-1]
nuvars = weights.shape[0]
nvconstraints = nsamples * nuvars
num_opt_vars = nbasis + nuvars + nvconstraints + 1
num_constraints = 2 * nsamples * nuvars + nsamples
v_coef = weights / (1 - beta) * 1. / nsamples * 1 / (1 - alpha)
#print (beta
#print (weights
#print (v_coef
Iquad = np.identity(nuvars)
Iv = np.identity(nvconstraints)
# num_quad_point = mu
# nsamples = nu
# nbasis = m
# design vars [c_1,...,c_m,u_1,...,u_{mu+1},v_1,...,v_{mu+1}nu]
# v_ij variables ordering: loop through j fastest, e.g. v_11,v_{12} etc
if not trapezoid_rule:
c_arr = np.hstack(
(basis_matrix.sum(axis=0) / nsamples, 1 / (1 - alpha) * weights,
np.repeat(v_coef, nsamples)))
# # v_ij+h'c+u_i <=y_j
# constraints_1 = np.hstack((
# -np.tile(basis_matrix,(nuvars,1)),
# -np.repeat(Iquad,nsamples,axis=0),-Iv))
# # v_ij >=0
# constraints_3 = np.hstack((
# np.zeros((nvconstraints,nbasis+nuvars)),-Iv))
# G_arr = np.vstack((constraints_1,constraints_3))
# assert G_arr.shape[0]==num_constraints
# assert G_arr.shape[1]==num_opt_vars
# assert c_arr.shape[0]==num_opt_vars
# I,J,data = sparse.find(G_arr)
# G = spmatrix(data,I,J,size=G_arr.shape)
#v_ij+h'c+u_i <=y_j
constraints_1_shape = (nvconstraints, num_opt_vars)
constraints_1a_I = np.repeat(np.arange(nvconstraints), nbasis)
constraints_1a_J = np.tile(np.arange(nbasis), nvconstraints)
constraints_1a_data = -np.tile(basis_matrix, (nquad_intervals + 1, 1))
constraints_1b_I = np.arange(nvconstraints)
constraints_1b_J = np.repeat(np.arange(nquad_intervals + 1),
nsamples) + nbasis
constraints_1b_data = -np.repeat(np.ones(nquad_intervals + 1),
nsamples)
ii = nbasis + nquad_intervals + 1
jj = ii + nvconstraints
constraints_1c_I = np.arange(nvconstraints)
constraints_1c_J = np.arange(ii, jj)
constraints_1c_data = -np.ones((nquad_intervals + 1) * nsamples)
constraints_1_data = np.hstack(
(constraints_1a_data.flatten(), constraints_1b_data,
constraints_1c_data))
constraints_1_I = np.hstack(
(constraints_1a_I, constraints_1b_I, constraints_1c_I))
constraints_1_J = np.hstack(
(constraints_1a_J, constraints_1b_J, constraints_1c_J))
# v_ij >=0
constraints_3_I = np.arange(constraints_1_shape[0],
constraints_1_shape[0] + nvconstraints)
constraints_3_J = np.arange(
nbasis + nquad_intervals + 1,
nbasis + nquad_intervals + 1 + nvconstraints)
constraints_3_data = -np.ones(nvconstraints)
constraints_shape = (num_constraints, num_opt_vars)
constraints_I = np.hstack((constraints_1_I, constraints_3_I))
constraints_J = np.hstack((constraints_1_J, constraints_3_J))
constraints_data = np.hstack((constraints_1_data, constraints_3_data))
G = spmatrix(constraints_data,
constraints_I,
constraints_J,
size=constraints_shape)
# assert np.allclose(np.asarray(matrix(G)),G_arr)
# print (constraints_shape
# print (np.asarray(matrix(G))
h_arr = np.hstack((-np.tile(values, nuvars), np.zeros(nvconstraints)))
#print (G_arr
#print (c_arr
#print (h_arr
else:
c_arr = np.hstack(
(basis_matrix.sum(axis=0) / nsamples, 1 / (1 - alpha) * weights,
np.repeat(v_coef,
nsamples), 1 / (nsamples * (1 - alpha)) * np.ones(1)))
# v_ij+h'c+u_i <=y_j
constraints_1 = np.hstack((-np.tile(basis_matrix,
(nquad_intervals, 1)),
-np.repeat(Iquad, nsamples, axis=0), -Iv,
np.zeros((nvconstraints, 1))))
#W+h'c<=y_j
constraints_2 = np.hstack((-basis_matrix, np.zeros(
(nsamples, nuvars)), np.zeros((nsamples, nvconstraints)), -np.ones(
(nsamples, 1))))
# v_ij >=0
constraints_3 = np.hstack((np.zeros(
(nvconstraints, nbasis + nquad_intervals)), -Iv,
np.zeros((nvconstraints, 1))))
G_arr = np.vstack((constraints_1, constraints_2, constraints_3))
h_arr = np.hstack(
(-np.tile(values, nuvars), -values, np.zeros(nvconstraints)))
assert G_arr.shape[0] == num_constraints
assert G_arr.shape[1] == num_opt_vars
assert c_arr.shape[0] == num_opt_vars
I, J, data = sparse.find(G_arr)
G = spmatrix(data, I, J, size=G_arr.shape)
c = matrix(c_arr)
h = matrix(h_arr)
if verbosity < 1:
solvers.options['show_progress'] = False
else:
solvers.options['show_progress'] = True
# solvers.options['abstol'] = 1e-10
# solvers.options['reltol'] = 1e-10
# solvers.options['feastol'] = 1e-10
sol = np.asarray(solvers.lp(c=c, G=G, h=h,
solver=solver_name)['x'])[:nbasis]
residuals = values - basis_matrix.dot(sol)[:, 0]
coef = np.append(conditional_value_at_risk(residuals, alpha), sol)
return coef
from cvxopt import matrix, solvers
c = matrix([-5., -3.]) # since we need to maximize the objective funtion
G = matrix([[1., 2., 1., -1., 0.], [1., 1., 4., 0., -1.]])
h = matrix([10., 16., 32., 0., 0.])
solvers.options['show_progress'] = False
sol = solvers.lp(c, G, h)
print('Solution"')
print(sol['x'])
def prob4():
"""Solve the allocation model problem in 'ForestData.npy'.
Note that the first three rows of the data correspond to the first
analysis area, the second group of three rows correspond to the second
analysis area, and so on.
Returns (in order):
The optimizer (sol['x'])
The optimal value (sol['primal objective']*-1000)
"""
data = np.load("ForestData.npy")
c = -data[:, 3]
t = -data[:, 4]
g = -data[:, 5]
w = -data[:, 6]
acres = data[::3, 1]
K = np.array([[
1., 1., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0.
],
[
0., 0., 0., 1., 1., 1., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 1., 1., 1., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 1., 1., 0., 0.,
0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 1.,
1., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 1., 1., 1., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 1., 1., 1.
],
[
-1., -1., -1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., -1., -1., -1., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., -1., -1., -1., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., -1., -1., -1., 0.,
0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -1., -1.,
-1., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., -1., -1., -1., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., -1., -1., -1.
]])
a = -np.eye(21)
G = np.vstack((K, t, g, w, a))
#G = np.vstack((K,t))
#G = np.vstack((G,g))
#G = np.vstack((G,w))
#G = np.vstack((G,a))
h = np.array([
acres[0], acres[1], acres[2], acres[3], acres[4], acres[5], acres[6],
-acres[0], -acres[1], -acres[2], -acres[3], -acres[4], -acres[5],
-acres[6], -40000., -5., (-70.0 * 788)
])
h = np.hstack((h, np.zeros(21)))
G = matrix(G)
h = matrix(h)
sol = solvers.lp(matrix(c), G, h)
return sol['x'], -1000 * sol['primal objective']
def compute_reward(n_states, n_actions, trans_prob, policy, discount,
max_reward, l1_reg):
"""
Find the reward function with linear programming IRL
n_states: number of states
n_actions: number of actions
trans_prob: transition probability which is a numpy array which maps
state(t), action(t) to state(t+1) (state_t, action_t, state_tp1)
policy: mapping of states to actions
discount: discount factor
max_reward: maximum reward
l1_reg: l1 regularization
return: reward vector
"""
A = set(range(n_actions))
trans_prob = np.transpose(trans_prob, (1, 0, 2))
# (P_a1(i) - P_a(i)) * inv(I - gamma * P_a1(i))
def T(a, s):
return np.dot(
trans_prob[policy[s], s] - trans_prob[a, s],
np.linalg.inv(np.eye(n_states) - discount * trans_prob[policy[s]]))
c = -np.hstack(
[np.zeros(n_states),
np.ones(n_states), -l1_reg * np.ones(n_states)])
zero_stack1 = np.zeros((n_states * (n_actions - 1), n_states))
T_stack = np.vstack([
-T(a, s) for s in range(n_states)
for a in A - {policy[s]} # other than the chosen a
])
I_stack1 = np.vstack([
np.eye(1, n_states, s) for s in range(n_states)
for a in A - {policy[s]}
])
I_stack2 = np.eye(n_states)
zero_stack2 = np.zeros((n_states, n_states))
D_left = np.vstack([T_stack, T_stack, -I_stack2, I_stack2])
D_middle = np.vstack([I_stack1, zero_stack1, zero_stack2, zero_stack2])
D_right = np.vstack([zero_stack1, zero_stack1, -I_stack2, -I_stack2])
D = np.hstack([D_left, D_middle, D_right])
b = np.zeros((n_states * (n_actions - 1) * 2 + 2 * n_states, 1))
bounds = np.array([(None, None)] * 2 * n_states +
[(-max_reward, max_reward)] * n_states)
D_bounds = np.hstack([
np.vstack([-np.eye(n_states), np.eye(n_states)]),
np.vstack(
[np.zeros((n_states, n_states)),
np.zeros((n_states, n_states))]),
np.vstack(
[np.zeros((n_states, n_states)),
np.zeros((n_states, n_states))])
])
b_bounds = np.vstack([max_reward * np.ones((n_states, 1))] * 2)
D = np.vstack((D, D_bounds))
b = np.vstack((b, b_bounds))
A_ub = matrix(D)
b = matrix(b)
c = matrix(c)
results = solvers.lp(c, A_ub, b)
r = np.asarray(results["x"][:n_states], dtype=np.double)
return r.reshape((n_states))
# maximize -h'*z - b'*w
# subject to +/- c + G'*z + A'*w >= 0
# z >= 0
#
# with variables z (8), w (1).
cc = matrix(0.0, (9,1))
cc[:8], cc[8] = h, b
GG = spmatrix([], [], [], (n+8, 9))
GG[:n,:8] = -G.T
GG[:n,8] = -A.T
GG[n::n+9] = -1.0
hh = matrix(0.0, (n+8,n))
hh[:n,:] = matrix([i>=j for i in range(n) for j in range(n)],
(n,n), 'd') # upper triangular matrix of ones
l = [-blas.dot(cc, solvers.lp(cc, GG, hh[:,k])['x']) for k in range(n)]
u = [blas.dot(cc, solvers.lp(cc, GG, -hh[:,k])['x']) for k in range(n)]
def f(x,y): return x+2*[y]
def stepl(x): return functools.reduce(f, x[1:], [x[0]])
def stepr(x): return functools.reduce(f, x[:-1], []) + [x[-1]]
try: import pylab
except ImportError: pass
else:
pylab.figure(1, facecolor='w')
pylab.plot(stepl(a), stepr(p), '-')
pylab.title('Maximum entropy distribution (fig. 7.2)')
pylab.xlabel('a')
pylab.ylabel('p = Prob(X = a)')
def zonotope_sampler(A_zono, **params):
""" MCMC based sampler for projection DPPs.
The similarity matrix is the orthogonal projection matrix onto
the row span of the feature vector matrix.
Samples are of size equal to the ransampl_size of the projection matrix
also equal to the rank of the feature matrix (assumed to be full row rank).
:param A_zono:
Feature vector matrix, feature vectors are stacked columnwise.
It is assumed to be full row rank.
:type A_zono:
array_like
:param params: Dictionary containing the parameters
- ``'lin_obj'`` (list): Linear objective (:math:`c`) of the linear program used to identify the tile in which a point lies. Default is a random Gaussian vector.
- ``'x_0'` (list): Initial point.
- ``'nb_iter'`` (int): Number of iterations of the MCMC chain. Default is 10.
- ``'T_max'`` (float): Maximum running time of the algorithm (in seconds).
Default is 10s.
:type params: dict
:return:
MCMC chain of approximate samples (stacked row_wise i.e. nb_iter rows).
:rtype:
array_like
.. seealso::
Algorithm 5 in :cite:`GaBaVa17`
- :func:`extract_basis <extract_basis>`
- :func:`basis_exchange_sampler <basis_exchange_sampler>`
"""
r, N = A_zono.shape # Sizes of r=samples=rank(A_zono), N=ground set
# Linear objective
c = matrix(params.get('lin_obj', np.random.randn(N)))
# Initial point x0 = A*u, u~U[0,1]^n
x0 = matrix(params.get('x_0', A_zono.dot(np.random.rand(N))))
nb_iter = params.get('nb_iter', 10)
T_max = params.get('T_max', None)
###################
# Linear problems #
###################
# Canonical form
# min c.T*x min c.T*x
# s.t. G*x <= h <=> s.t. G*x + s = h
# A*x = b A*x = b
# s >= 0
# CVXOPT
# =====> solvers.lp(c, G, h, A, b, solver='glpk')
#################################################
# To access the tile Z(B_x)
# Solve P_x(A,c)
######################################################
# y^* =
# argmin c.T*y argmin c.T*y
# s.t. A*y = x <=> s.t. A *y = x
# 0 <= y <= 1 [ I_n] *y <= [1^n]
# [-I_n] [0^n]
######################################################
# Then B_x = \{ i ; y_i^* \in ]0,1[ \}
A = spmatrix(0.0, [], [], (r, N))
A[:, :] = A_zono
G = spmatrix(0.0, [], [], (2 * N, N))
G[:N, :] = spmatrix(1.0, range(N), range(N))
G[N:, :] = spmatrix(-1.0, range(N), range(N))
# Endpoints of segment
# D_x \cap Z(A) = [x+alpha_m*d, x-alpha_M*d]
###########################################################################
# alpha_m/_M = argmin +/-alpha argmin [+/-1 0^N].T * [alpha,lambda]
# s.t. x + alpha d = A lambda <=> s.t. [-d A] *[alpha, lambda] = x
# 0 <= lambda <= 1 [0^N I_N] *[alpha, lambda] <= [1^N]
# [0^N -I_N] [0^N]
##########################################################################
c_mM = matrix(0.0, (N + 1, 1))
c_mM[0] = 1.0
A_mM = spmatrix(0.0, [], [], (r, N + 1))
A_mM[:, 1:] = A
G_mM = spmatrix(0.0, [], [], (2 * N, N + 1))
G_mM[:, 1:] = G
# Common h to both kind of LP
# cf. 0 <= y <= 1 and 0 <= lambda <= 1
h = matrix(0.0, (2 * N, 1))
h[:N, :] = 1.0
##################
# Initialization #
##################
B_x0 = []
while len(B_x0) != r:
# Initial tile B_x0
# Solve P_x0(A,c)
y_star = solvers.lp(c, G, h, A, x0, solver='glpk')['x']
# Get the tile
B_x0 = extract_basis(np.asarray(y_star))
# Initialize sequence of sample
Bases = [B_x0]
# Compute the det of the tile (Vol(B)=abs(det(B)))
det_B_x0 = la.det(A_zono[:, B_x0])
it, t_start = 1, time.time()
flag = it < nb_iter
while flag:
# Take uniform direction d defining D_x0
d = matrix(np.random.randn(r, 1))
# Define D_x0 \cap Z(A) = [x0 + alpha_m*d, x0 - alpha_M*d]
# Update the constraint [-d A] * [alpha,lambda] = x
A_mM[:, 0] = -d
# Find alpha_m/M
alpha_m = solvers.lp(c_mM, G_mM, h, A_mM, x0, solver='glpk')['x'][0]
alpha_M = solvers.lp(-c_mM, G_mM, h, A_mM, x0, solver='glpk')['x'][0]
# Propose x1 ~ U_{[x0+alpha_m*d, x0-alpha_M*d]}
x1 = x0 + (alpha_m + (alpha_M - alpha_m) * np.random.rand()) * d
# Proposed tile B_x1
# Solve P_x1(A,c)
y_star = solvers.lp(c, G, h, A, x1, solver='glpk')['x']
# Get the tile
B_x1 = extract_basis(np.asarray(y_star))
# Accept/Reject the move with proba Vol(B1)/Vol(B0)
if len(B_x1) != r: # if extract_basis returned smtg ill conditioned
Bases.append(B_x0)
else:
det_B_x1 = la.det(A_zono[:, B_x1])
if np.random.rand() < abs(det_B_x1 / det_B_x0):
x0, B_x0, det_B_x0 = x1, B_x1, det_B_x1
Bases.append(B_x1)
else:
Bases.append(B_x0)
it += 1
flag = (it < nb_iter) if not T_max else ((time.time()-t_start) < T_max)
return np.array(Bases)
def emd(D1, D2, gdist, scale=False):
"""
Compute the Earth Mover's Distance (EMD) between two sets of elements,
here dictionary atoms, using a ground distance.
Possible choice are "chordal", "fubinistudy", "binetcauchy", "geodesic",
"frobenius", "abs_euclidean" or "euclidean".
The scale parameter changes the return value to be between 0 and 1.
"""
g = _valid_atom_metric(gdist)
if g is None:
print("Unknown ground distance, exiting.")
return NaN
# if gdist == "chordal":
# g = chordal
# elif gdist == "chordal_principal_angles":
# g = chordal_principal_angles
# elif gdist == "fubinistudy":
# g = fubini_study
# elif gdist == "binetcauchy":
# g = binet_cauchy
# elif gdist == "geodesic":
# g = geodesic
# elif gdist == "frobenius":
# g = frobenius
# elif gdist == "abs_euclidean":
# g = abs_euclidean
# elif gdist == "euclidean":
# g = euclidean
# else:
# print 'Unknown ground distance, exiting.'
# return NaN
# # Do we need a registration? If kernel do not have the same shape, yes
# if not np.all(np.array([i.shape[0] for i in D1+D2]) == D1[0].shape[0]):
# # compute correlation and distance matrices
# k_dim = D1[0].shape[1]
# # minl = np.array([i.shape[1] for i in D1+D2]).min()
# max_l1 = np.array([i.shape[0] for i in D1]).max()
# max_l2 = np.array([i.shape[0] for i in D2]).max()
# if max_l2 > max_l1:
# Da = D1
# Db = D2
# max_l = max_l2
# else:
# Da = D2
# Db = D1
# max_l = max_l1
# Dbe = []
# for i in range(len(Db)):
# k_l = Db[i].shape[0]
# Dbe.append(np.concatenate((zeros((max_l-k_l, k_dim)), Db[i]), axis=0))
# gdm = zeros((len(Da), len(Db)))
# for i in range(len(Da)):
# k_l = Da[i].shape[0]
# m_dist, m_corr = _kernel_registration(np.concatenate((zeros(( np.int(np.floor((max_l-k_l)/2.)), k_dim)), Da[i], zeros((np.int(np.ceil((max_l-k_l)/2.)), k_dim))), axis=0), Dbe, g)
# for j in range(len(Dbe)):
# gdm[i,j] = m_dist[j, np.unravel_index(np.abs(m_corr[j,:]).argmax(), m_corr[j,:].shape)]
# else:
# # all atoms have the same length, no registration
# gdm = np.zeros((len(D1), len(D2)))
# for i in range(len(D1)):
# for j in range(len(D2)):
# gdm[i,j] = g(D1[i], D2[j])
gdm = _compute_gdm(D1, D2, g)
c = co.matrix(gdm.flatten(order="F"))
G1 = co.spmatrix([], [], [], (len(D1), len(D1) * len(D2)))
G2 = co.spmatrix([], [], [], (len(D2), len(D1) * len(D2)))
G3 = co.spmatrix(-1.0, range(len(D1) * len(D2)), range(len(D1) * len(D2)))
for i in range(len(D1)):
for j in range(len(D2)):
k = j + (i * len(D2))
G1[i, k] = 1.0
G2[j, k] = 1.0
G = co.sparse([G1, G2, G3])
h1 = co.matrix(1.0 / len(D1), (len(D1), 1))
h2 = co.matrix(1.0 / len(D2), (len(D2), 1))
h3 = co.spmatrix([], [], [], (len(D1) * len(D2), 1))
h = co.matrix([h1, h2, h3])
A = co.matrix(1.0, (1, len(D1) * len(D2)))
b = co.matrix([1.0])
co.solvers.options["show_progress"] = False
sol = solv.lp(c, G, h, A, b)
d = sol["primal objective"]
if not scale:
return d
else:
return _scale_metric(gdist, d, D1)
def findMinQ2(Q1):
c = matrix(
[
-Q1[0,0],
-Q1[0, 1],
-Q1[0, 2],
-Q1[0, 3],
-Q1[0, 4],
-Q1[1, 0],
-Q1[1, 1],
-Q1[1, 2],
-Q1[1, 3],
-Q1[1, 4],
-Q1[2, 0],
-Q1[2, 1],
-Q1[2, 2],
-Q1[2, 3],
-Q1[2, 4],
-Q1[3, 0],
-Q1[3, 1],
-Q1[3, 2],
-Q1[3, 3],
-Q1[3, 4],
-Q1[3, 0],
-Q1[4, 1],
-Q1[4, 2],
-Q1[4, 3],
-Q1[4, 4],
-Q2[0, 0],
-Q2[0, 1],
-Q2[0, 2],
-Q2[0, 3],
-Q2[0, 4],
-Q2[1, 0],
-Q2[1, 1],
-Q2[1, 2],
-Q2[1, 3],
-Q2[1, 4],
-Q2[2, 0],
-Q2[2, 1],
-Q2[2, 2],
-Q2[2, 3],
-Q2[2, 4],
-Q2[3, 0],
-Q2[3, 1],
-Q2[3, 2],
-Q2[3, 3],
-Q2[3, 4],
-Q2[3, 0],
-Q2[4, 1],
-Q2[4, 2],
-Q2[4, 3],
-Q2[4, 4]
]
)
# c = matrix([
# -Q1[0],-Q1[1],-Q1[2],-Q1[3],-Q1[4],-Q1[5],-Q1[6],-Q1[7],-Q1[8],-Q1[9],
# -Q1[10],-Q1[11],-Q1[12],-Q1[13],-Q1[14],-Q1[15],-Q1[16],-Q1[17],-Q1[18],-Q1[19],
# -Q1[20],-Q1[21],-Q1[22],-Q1[23],-Q1[24]
# ])
G = matrix(np.identity(50)*-1)
h = matrix([ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0])
A = matrix([
[1.],[1.],[1.],[1.],[1.],[1.],[1.],[1.],[1.],[1.],
[1.], [1.], [1.], [1.], [1.], [1.], [1.], [1.], [1.], [1.],
[1.], [1.], [1.], [1.], [1.],
[1.], [1.], [1.], [1.], [1.], [1.], [1.], [1.], [1.], [1.],
[1.], [1.], [1.], [1.], [1.], [1.], [1.], [1.], [1.], [1.],
[1.], [1.], [1.], [1.], [1.]
])
b = matrix([1.])
solvers.options['show_progress'] = False
sol = solvers.lp(c, G, h, A, b)
primeDist = [sol['x'][0],sol['x'][1],sol['x'][2],sol['x'][3],sol['x'][4],sol['x'][5],sol['x'][6],sol['x'][7],sol['x'][8],sol['x'][9],
sol['x'][10], sol['x'][11], sol['x'][12], sol['x'][13], sol['x'][14], sol['x'][15], sol['x'][16], sol['x'][17], sol['x'][18], sol['x'][19],
sol['x'][20], sol['x'][21], sol['x'][22], sol['x'][23], sol['x'][24],
#
# sol['x'][25], sol['x'][26], sol['x'][27], sol['x'][28], sol['x'][29], sol['x'][30], sol['x'][31], sol['x'][32],
# sol['x'][33], sol['x'][34],
# sol['x'][35], sol['x'][36], sol['x'][37], sol['x'][38], sol['x'][39], sol['x'][40], sol['x'][41],
# sol['x'][42], sol['x'][43], sol['x'][44],
# sol['x'][45], sol['x'][46], sol['x'][47], sol['x'][48], sol['x'][49]
]
return primeDist