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ADMM_slack.py
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ADMM_slack.py
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# Source: https://github.com/cvxgrp/lfd_lqr/blob/master/algorithms.py
import warnings
import numpy as np
import cvxpy as cp
from scipy.linalg import solve_discrete_are
import matplotlib.pyplot as plt
def _ADMM(L, LPQR, r, rPQR, A, B, P, Q, R, niter=50, rho=1, plot=False):
"""
Policy fitting with a Kalman constraint.
Args:
- L: function that takes in a cvxpy Variable
and returns a cvxpy expression representing the objective.
- r: function that takes in a cvxpy Variable
and returns a cvxpy expression and a list of constraints
representing the regularization function.
- xs: N x n matrix of states.
- us_observed: N x m matrix of inputs.
- A: n x n dynamics matrix.
- B: n x m dynamics matrix.
- P: n x n PSD matrix, the initial PSD cost-to-go coefficient.
- Q: n x n PSD matrix, the initial state cost coefficient.
- R: n x n PD matrix, the initial input cost coefficient.
- niter: int (optional). Number of iterations (default=50).
- rho: double (optional). Penalty parameter (default=1).
Returns:
- K: m x n gain matrix found by policy fitting with a Kalman constraint.
"""
n, m = B.shape
K = np.zeros((m, n))
Y = np.zeros((n + m, n))
s = np.zeros((m, n))
history = {k: [] for k in 'KPQRYs'}
try:
import mosek
solver = cp.MOSEK
except:
print("Solver MOSEK is not installed, falling back to SCS.", flush=True)
solver = cp.SCS
losses = []
for k in range(niter):
# K step
Kcp = cp.Variable((m, n))
M = cp.vstack([
Q + A.T@P@(A + B@Kcp) - P,
R@Kcp + B.T@P@(A + B@Kcp) - s
])
objective = cp.Minimize(L(Kcp) + r(Kcp) + cp.trace(Y.T@M) + rho/2*cp.sum_squares(M))
prob = cp.Problem(objective, r(Kcp))
try:
prob.solve(solver=solver)
except:
try:
print("Defaulting to SCS solver for PQR step", flush=True)
prob.solve(solver=cp.SCS, acceleration_lookback=0, max_iters=10000)
except:
print("SCS solver failed", flush=True)
Kinf = np.inf*np.ones((m, n))
Pinf = np.inf*np.ones((n, n))
Qinf = np.inf*np.ones((n, n))
Rinf = np.inf*np.ones((m, m))
return Kinf, Pinf, Qinf, Rinf
K = Kcp.value
# R step
Rcp = cp.Variable((m, m), PSD=True)
M = cp.vstack([
Q + A.T@P@(A + B@K) - P,
Rcp@K + B.T@P@(B@K + A) - s
])
objective = cp.Minimize(LPQR(Q, Rcp) + rPQR(Q, Rcp) + cp.trace(Y.T@M) +
rho/2*cp.sum_squares(M))
prob = cp.Problem(objective, [Rcp>>np.eye(m)])
try:
prob.solve(solver=solver)
except:
try:
print("Defaulting to SCS solver for PQR step", flush=True)
prob.solve(solver=cp.SCS, acceleration_lookback=0, max_iters=10000)
except:
print("SCS solver failed", flush=True)
Kinf = np.inf*np.ones((m, n))
Pinf = np.inf*np.ones((n, n))
Qinf = np.inf*np.ones((n, n))
Rinf = np.inf*np.ones((m, m))
return Kinf, Pinf, Qinf, Rinf
R = Rcp.value
# P, Q, R, s step
Pcp = cp.Variable((n, n), PSD=True)
Qcp = cp.Variable((n, n), PSD=True)
# Rcp = cp.Variable((m, m), PSD=True)
scp = cp.Variable((m, n))
M = cp.vstack([
Qcp + A.T@Pcp@(A + B@K) - Pcp,
R@K + B.T@Pcp@(B@K + A) - scp
])
# M = cp.vstack([
# Qcp + A.T@Pcp@(A + B@K) - Pcp,
# Rcp@K + B.T@Pcp@(B@K + A) - scp
# ])
objective = cp.Minimize(LPQR(Qcp, R) + rPQR(Qcp, R) + 20*r(scp) + cp.trace(Y.T@M) +
rho/2*cp.sum_squares(M))
prob = cp.Problem(objective, [Pcp>>0, Qcp>>0])
# objective = cp.Minimize(LPQR(Qcp, Rcp) + rPQR(Qcp, Rcp) + r(scp) + cp.trace(Y.T@M) +
# rho/2*cp.sum_squares(M))
# prob = cp.Problem(objective, [Pcp>>0, Qcp>>0, Rcp>>np.eye(m)])
try:
prob.solve(solver=solver)
except:
try:
print("Defaulting to SCS solver for PQR step", flush=True)
prob.solve(solver=cp.SCS, acceleration_lookback=0, max_iters=10000)
except:
print("SCS solver failed", flush=True)
Kinf = np.inf*np.ones((m, n))
Pinf = np.inf*np.ones((n, n))
Qinf = np.inf*np.ones((n, n))
Rinf = np.inf*np.ones((m, m))
return Kinf, Pinf, Qinf, Rinf
P = Pcp.value
Q = Qcp.value
# R = Rcp.value
s = scp.value
# print('Inside ADMM:', np.shape(K), np.shape(P), np.shape(Q), np.shape(R), np.shape(A), np.shape(B), flush=True)
# Y step
residual = np.vstack([
Q + A.T@P@(A + B@K) - P,
R@K + B.T@P@(A + B@K) - s
])
Y = Y + rho*residual
M = cp.vstack([
Q + A.T@P@(A + B@K) - P,
R@K + B.T@P@(B@K + A) - s
])
# print(Y)
# print(M)
# print(cp.trace(Y.T@M).value)
losses.append(L(K).value + LPQR(Q, R).value + r(K).value + rPQR(Q, R).value + r(s).value + cp.trace(
Y.T@M).value + rho/2*cp.sum_squares(M).value)
history['K'].append(K)
history['P'].append(P)
history['Q'].append(Q)
history['R'].append(R)
history['s'].append(s)
# print(L(K).value)
R = (R + R.T)/2
Q = (Q + Q.T)/2
w, v = np.linalg.eigh(R)
w[w < 1e-6] = 1e-6
R = [email protected](w)@v.T
w, v = np.linalg.eigh(Q)
w[w < 0] = 0
Q = [email protected](w)@v.T
P = solve_discrete_are(A, B, Q, R)
if plot:
fig = plt.figure()
plt.plot(range(len(losses)), losses)
plt.xlabel("Iteration #")
plt.ylabel("ADMM Loss")
plt.title("ADMM Loss vs. Iterations")
plt.grid(True)
for k in history:
print(k)
print(history[k])
print()
plt.show()
return -np.linalg.solve(R + B.T@P@B, B.T@P@A), P, Q, R
def policy_fitting(L, r, n, m):
"""
Traditional policy fitting (no ADMM)
:param L: L(K), Loss function
:param r: r(K), regularization term
:param xs: Array of observed states (N x n)
:param us: Array of observed inputs (N x m)
:return: Kcp (gain matrix found by policy fitting) (m x n)
"""
Kpf = cp.Variable((m, n))
r_obj = r(Kpf)
cp.Problem(cp.Minimize(L(Kpf) + r_obj)).solve()
return Kpf.value
def policy_fitting_with_a_kalman_constraint(L, r, A, B, n_random=5, niter=50, rho=10,
P0=None, Q0=None, R0=None, LPQR=None, rPQR=None):
"""
Wrapper around _ADMM.
"""
n, m = B.shape
def evaluate_L(K):
Kcp = cp.Variable((m, n))
Kcp.value = K
loss = L(K)
return loss.value
if LPQR is None:
LPQR = lambda Q, R: cp.Constant(0)
if rPQR is None:
rPQR = lambda Q, R: cp.Constant(0)
# solve with zero initialization
P = np.zeros((n, n))
Q = np.zeros((n, n))
R = np.zeros((m, m))
K, P, Q, R = _ADMM(L, LPQR, r, rPQR, A, B, P, Q, R, niter=niter, rho=rho, plot=False)
best_K, bP, bQ, bR = K, P, Q, R
best_L = evaluate_L(K)
if P0 is not None and R0 is not None and Q0 is not None:
K, P, Q, R = _ADMM(L, LPQR, r, rPQR, A, B, P0, Q0, R0, niter=niter, rho=rho)
L_K = evaluate_L(K)
if L_K < best_L:
best_L = L_K
best_K, bP, bQ, bR = K, P, Q, R
# run n_random random initializations; keep best
for iter in range(n_random):
P = 1./np.sqrt(n)*np.random.randn(n, n)
Q = 1./np.sqrt(n)*np.random.randn(n, n)
R = 1./np.sqrt(m)*np.random.randn(m, m)
P = P.T@P
Q = Q.T@Q
R = R.T@R
K, P, Q, R = _ADMM(L, LPQR, r, rPQR, A, B, P, Q, R,
niter=niter, rho=rho)
L_K = evaluate_L(K)
if L_K < best_L:
best_L = L_K
best_K, bP, bQ, bR = K, P, Q, R
return best_K, bP, bQ, bR