def solve_SAA(params, Xi, r):
N = Xi.shape[0]
Xi_pic = [pic.new_param('xi_' + str(i + 1), Xi[i]) for i in range(N)]
# Unpack problem parameters
c, q, xi_0, b, B = unpack_params_picos(params)
r = pic.new_param('r', r)
# Unpack artificial parameters
E, c_tilde, E_tilde_arr, E_arr, e5 = define_art_params(
c.get_value(), q.get_value(), xi_0)
E, c_tilde, E_tilde_arr, E_arr, e5 = convert_art_params_picos(
E, c_tilde, E_tilde_arr, E_arr, e5)
# Get dimensions.
x_dim = len(c)
xi_dim = len(xi_0)
num_piecewise_lin_f = len(E_tilde_arr)
# Initialize picos problem.
pb = pic.Problem()
# Add variables.
x_tilde = pb.add_variable("x_tilde", x_dim + 1)
t = pb.add_variable("t", N)
eta1_arr = [
pb.add_variable("eta1_" + str(i + 1), xi_dim) for i in range(2)
]
eta2_arr = [pb.add_variable("eta2_" + str(i + 1), 10) for i in range(2)]
# Add constraints.
for i in range(N):
for j in range(num_piecewise_lin_f):
pb.add_constraint((E_tilde_arr[j] * Xi_pic[i]) | x_tilde <= t[i])
for j in range(2):
pb.add_constraint(
abs(B.T * E_arr[j].T * x_tilde - B.T *
(eta1_arr[j] - eta2_arr[j])) <= MAX_MAN_HOUR -
(x_tilde.T * E_arr[j] * xi_0 / r) -
((eta1_arr[j] + eta2_arr[j]) | b / r))
# positivity of lagrange multipliers
pb.add_constraint(eta1_arr[j] >= 0)
pb.add_constraint(eta2_arr[j] >= 0)
pb.add_constraint(E.T * x_tilde >= 0)
pb.add_constraint(e5 | x_tilde == -1)
# Set objective.
pb.set_objective("min", (1 / N) * (1 | t) - (c_tilde | x_tilde))
solution = pb.solve(solver='cvxopt')
opt_val = -pb.obj_value()
opt_sol = np.array(solution['cvxopt_sol']['x'][:x_dim]).reshape(-1).round(
5) # round 5 digits
return opt_sol, opt_val
def solve_moment_based_partial_lifting(params, r, mu, t, g):
g = pic.new_param('g', g)
# Unpack problem parameters
c, q, xi_0, b, B = unpack_params_picos(params)
r = pic.new_param('r', r)
# Unpack artificial parameters
E, c_tilde, E_tilde_arr, E_arr, e5 = define_art_params(
c.get_value(), q.get_value(), xi_0)
E, c_tilde, E_tilde_arr, E_arr, e5 = convert_art_params_picos(
E, c_tilde, E_tilde_arr, E_arr, e5)
# Get dimensions.
x_dim = len(c)
xi_dim = len(xi_0)
num_piecewise_lin_f = len(E_tilde_arr)
t_dim = len(t)
# Initialize picos problem.
pb = pic.Problem()
# Add variables.
x_tilde = pb.add_variable("x_tilde", x_dim + 1)
kappa = pb.add_variable("kappa", 1)
beta = pb.add_variable("beta", xi_dim)
delta = pb.add_variable("delta", t_dim)
alpha_arr = []
for i in range(t_dim):
for j in range(num_piecewise_lin_f):
alpha_arr.append(
pb.add_variable("alpha_" + str(i + 1) + "_" + str(j + 1), 3))
lambda1_arr = [
pb.add_variable("lambda1_" + str(i + 1), xi_dim)
for i in range(num_piecewise_lin_f)
]
lambda2_arr = [
pb.add_variable("lambda2_" + str(i + 1), xi_dim)
for i in range(num_piecewise_lin_f)
]
eta1_arr = [
pb.add_variable("eta1_" + str(i + 1), xi_dim) for i in range(2)
]
eta2_arr = [pb.add_variable("eta2_" + str(i + 1), 10) for i in range(2)]
# Add constraints.
for j in range(num_piecewise_lin_f):
pb.add_constraint((lambda1_arr[j] | (xi_0 + b)) -
(lambda2_arr[j] | (xi_0 - b)) + 0.5 *
(sum(alpha_arr[j::num_piecewise_lin_f])[2] -
sum(alpha_arr[j::num_piecewise_lin_f])[1]) <= kappa)
pb.add_constraint(0.5 *
(sum(alpha_arr[j::num_piecewise_lin_f])[2] +
sum(alpha_arr[j::num_piecewise_lin_f])[1]) == delta)
pb.add_constraint(lambda1_arr[j] - lambda2_arr[j] - sum([
alpha[0] * g[i, :].T for alpha, i in zip(
alpha_arr[j::num_piecewise_lin_f], range(t_dim))
]) + beta == E_tilde_arr[j].T * x_tilde)
# positivity of lagrange multipliers
pb.add_constraint(lambda1_arr[j] >= 0)
pb.add_constraint(lambda2_arr[j] >= 0)
# conic dual variables (alpha in L^3)
for alpha_ij in alpha_arr[j::num_piecewise_lin_f]:
pb.add_constraint(abs(alpha_ij[0] // alpha_ij[1]) <= alpha_ij[2])
for k in range(2):
pb.add_constraint(
abs(B.T * E_arr[k].T * x_tilde - B.T *
(eta1_arr[k] - eta2_arr[k])) <= MAX_MAN_HOUR -
(x_tilde.T * E_arr[k] * xi_0 / r) -
((eta1_arr[k] + eta2_arr[k]) | b / r))
# positivity of lagrange multipliers
pb.add_constraint(eta1_arr[k] >= 0)
pb.add_constraint(eta2_arr[k] >= 0)
pb.add_constraint(E.T * x_tilde >= 0)
pb.add_constraint(e5 | x_tilde == -1)
# Set objective.
pb.set_objective("min",
kappa + (beta | mu) + (delta | t) - (c_tilde | x_tilde))
solution = pb.solve(solver='cvxopt')
opt_val = -pb.obj_value()
opt_sol = np.array(
solution['cvxopt_sol']['x'][:x_dim]).reshape(-1).round(5)
return opt_sol, opt_val
def solve_moment_based(params, r, mu, Sigma):
# Unpack problem parameters
c, q, xi_0, b, B = unpack_params_picos(params)
r = pic.new_param('r', r)
# Unpack artificial parameters
E, c_tilde, E_tilde_arr, E_arr, e5 = define_art_params(
c.get_value(), q.get_value(), xi_0)
E, c_tilde, E_tilde_arr, E_arr, e5 = convert_art_params_picos(
E, c_tilde, E_tilde_arr, E_arr, e5)
# Get dimensions.
x_dim = len(c)
xi_dim = len(xi_0)
num_piecewise_lin_f = len(E_tilde_arr)
# Initialize picos problem.
pb = pic.Problem()
# Add variables.
x_tilde = pb.add_variable("x_tilde", x_dim + 1)
t = pb.add_variable("t", 1)
theta_arr = [
pb.add_variable("theta_" + str(i + 1), 1)
for i in range(num_piecewise_lin_f)
]
beta = pb.add_variable("beta", xi_dim)
phi_arr = [
pb.add_variable("phi_" + str(i + 1), xi_dim)
for i in range(num_piecewise_lin_f)
]
Lambda = pb.add_variable("Lambda", (xi_dim, xi_dim), vtype='symmetric')
lambda1_arr = [
pb.add_variable("lambda1_" + str(i + 1), xi_dim)
for i in range(num_piecewise_lin_f)
]
lambda2_arr = [
pb.add_variable("lambda2_" + str(i + 1), xi_dim)
for i in range(num_piecewise_lin_f)
]
eta1_arr = [
pb.add_variable("eta1_" + str(i + 1), xi_dim) for i in range(2)
]
eta2_arr = [
pb.add_variable("eta2_" + str(i + 1), xi_dim) for i in range(2)
]
# Add constraints.
for i in range(num_piecewise_lin_f):
pb.add_constraint((lambda1_arr[i] | (xi_0 + b)) -
(lambda2_arr[i] | (xi_0 - b)) -
(2 * phi_arr[i] | mu) + theta_arr[i] <= t)
pb.add_constraint(lambda1_arr[i] - lambda2_arr[i] - 2 * phi_arr[i] +
beta == E_tilde_arr[i].T * x_tilde)
pb.add_constraint((Lambda & phi_arr[i]) //
(phi_arr[i].T & theta_arr[i]) >> 0)
# positivity of lagrange multipliers
pb.add_constraint(lambda1_arr[i] >= 0)
pb.add_constraint(lambda2_arr[i] >= 0)
for j in range(2):
pb.add_constraint(
abs(B.T * E_arr[j].T * x_tilde - B.T *
(eta1_arr[j] - eta2_arr[j])) <= MAX_MAN_HOUR -
(x_tilde.T * E_arr[j] * xi_0 / r) -
((eta1_arr[j] + eta2_arr[j]) | b / r))
# positivity of lagrange multipliers
pb.add_constraint(eta1_arr[j] >= 0)
pb.add_constraint(eta2_arr[j] >= 0)
pb.add_constraint(E.T * x_tilde >= 0)
pb.add_constraint(e5 | x_tilde == -1)
# Set objective.
pb.set_objective("min",
t + (beta | mu) + (Lambda | Sigma) - (c_tilde | x_tilde))
solution = pb.solve(solver='cvxopt')
opt_val = -pb.obj_value()
opt_sol = np.array(
solution['cvxopt_sol']['x'][:x_dim]).reshape(-1).round(5)
return opt_sol, opt_val
def solve_SRO(params, Xi, eps_N):
'''
Xi is essential as input, since we need the values of the realizations rather
than some statistics of (e.g. mean, covariance, etc.)
'''
Xi_pic = pic.new_param('Xi', Xi)
eps_N = pic.new_param('eps_N', eps_N)
N = Xi.shape[0]
# Unpack problem parameters
c, q, xi_0, b, B = unpack_params_picos(params)
# Unpack artificial parameters
E, c_tilde, E_tilde_arr, E_arr, e5 = define_art_params(
c.get_value(), q.get_value(), xi_0)
E, c_tilde, E_tilde_arr, E_arr, e5 = convert_art_params_picos(
E, c_tilde, E_tilde_arr, E_arr, e5)
# Get dimensions.
x_dim = len(c)
xi_dim = len(xi_0)
num_piecewise_lin_f = len(E_tilde_arr)
# Initialize picos problem.
pb = pic.Problem()
# Add variables.
x_tilde = pb.add_variable("x_tilde", x_dim + 1)
t = pb.add_variable("t", N)
lambda_arr = []
lambda1_arr = []
lambda2_arr = []
eta_arr = []
eta1_arr = []
eta2_arr = []
for i in range(N):
for j in range(num_piecewise_lin_f):
lambda_arr.append(
pb.add_variable("lambda_" + str(i + 1) + "_" + str(j + 1), 1))
lambda1_arr.append(
pb.add_variable("lambda1_" + str(i + 1) + "_" + str(j + 1),
xi_dim))
lambda2_arr.append(
pb.add_variable("lambda2_" + str(i + 1) + "_" + str(j + 1),
xi_dim))
for k in range(2):
eta_arr.append(
pb.add_variable("eta_" + str(i + 1) + "_" + str(k + 1), 1))
eta1_arr.append(
pb.add_variable("eta1_" + str(i + 1) + "_" + str(k + 1),
xi_dim))
eta2_arr.append(
pb.add_variable("eta2_" + str(i + 1) + "_" + str(k + 1),
xi_dim))
# Add constraints.
for i in range(N):
Xi_i_hat = Xi_pic[i, :].T
for j in range(num_piecewise_lin_f):
pb.add_constraint(
((E_tilde_arr[j] * Xi_i_hat) | x_tilde) -
((Xi_i_hat - xi_0 - b) | lambda1_arr[
(num_piecewise_lin_f * i) + j]) +
((Xi_i_hat - xi_0 + b) | lambda2_arr[
(num_piecewise_lin_f * i) + j]) +
(lambda_arr[(num_piecewise_lin_f * i) + j] * eps_N) <= t[i])
pb.add_constraint(
abs((E_tilde_arr[j].T * x_tilde) -
lambda1_arr[(num_piecewise_lin_f * i) + j] +
lambda2_arr[(num_piecewise_lin_f * i) + j]) <= lambda_arr[
(num_piecewise_lin_f * i) + j])
# positivity of lagrange multipliers
pb.add_constraint(lambda_arr[(num_piecewise_lin_f * i) + j] >= 0)
pb.add_constraint(lambda1_arr[(num_piecewise_lin_f * i) + j] >= 0)
pb.add_constraint(lambda2_arr[(num_piecewise_lin_f * i) + j] >= 0)
for k in range(2):
pb.add_constraint(((E_arr[k] * Xi_i_hat) | x_tilde) -
((Xi_i_hat - xi_0 - b) | eta1_arr[(2 * i) + k]) +
((Xi_i_hat - xi_0 + b) | eta2_arr[(2 * i) + k]) +
(eta_arr[(2 * i) + k] * eps_N) <= MAX_MAN_HOUR)
pb.add_constraint(
abs((E_arr[k].T * x_tilde) - eta1_arr[(2 * i) + k] +
eta2_arr[(2 * i) + k]) <= eta_arr[(2 * i) + k])
# positivity of lagrange multipliers
pb.add_constraint(eta_arr[(2 * i) + k] >= 0)
pb.add_constraint(eta1_arr[(2 * i) + k] >= 0)
pb.add_constraint(eta2_arr[(2 * i) + k] >= 0)
pb.add_constraint(E.T * x_tilde >= 0)
pb.add_constraint(e5 | x_tilde == -1)
# Set objective.
pb.set_objective("min", (1 / N) * (1 | t) - (c_tilde | x_tilde))
solution = pb.solve(solver='cvxopt')
opt_val = -pb.obj_value()
opt_sol = np.array(
solution['cvxopt_sol']['x'][:x_dim]).reshape(-1).round(5)
return opt_sol, opt_val