policy_iteration_example.ipynb

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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy\n",
    "import scipy.linalg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "rng = np.random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n, d = 3, 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def spectral_radius(M):\n",
    "    return max(np.abs(np.linalg.eigvals(M)))\n",
    "\n",
    "\n",
    "def operator_norm(M):\n",
    "    return np.linalg.norm(M, ord=2)\n",
    "\n",
    "\n",
    "def random_contractive_system(n, d, rng):\n",
    "    B = rng.normal(size=(n, d))\n",
    "    \n",
    "    A = rng.normal(size=(n, n))\n",
    "    A /= operator_norm(A)\n",
    "    return 0.99 * A, B\n",
    "        \n",
    "\n",
    "def dlyap(A, Q):\n",
    "    # newer versions of scipy solve A P A^T - P + Q = 0,\n",
    "    # while older ones solve A^T P A - P + Q = 0. I do not\n",
    "    # remember exactly which version of scipy made the change.\n",
    "\n",
    "    # I am going to assume you have the newer version installed.\n",
    "    # If the assertion below fails, please add an if statement\n",
    "    # that branches on your version.\n",
    "\n",
    "    P = scipy.linalg.solve_discrete_lyapunov(A.T, Q)\n",
    "\n",
    "    assert np.allclose(A.T.dot(P).dot(A) - P, -Q)\n",
    "\n",
    "    return P\n",
    "    \n",
    "    \n",
    "def policy_iteration(A, B, K, S, R):\n",
    "    \n",
    "    A_clp = A + B @ K\n",
    "    n, d = B.shape\n",
    "    \n",
    "    assert spectral_radius(A_clp) < 1\n",
    "    \n",
    "    # first, solve for the value function\n",
    "    # V = dlyap(A+BK, S + K^T R K)\n",
    "    V = dlyap(A_clp, S + K.T @ R @ K)\n",
    "    \n",
    "    # form the Q matrix:\n",
    "    # Q = diag(S, R) + [A^T ; B^T] V [A B]\n",
    "    Q = scipy.linalg.block_diag(S, R) + np.vstack((A.T, B.T)) @ V @ np.hstack((A, B))\n",
    "    \n",
    "    # K_next = -Q_{22}^{-1} Q_{12}^T\n",
    "    return -scipy.linalg.solve(Q[n:, n:], Q[:n, n:].T, sym_pos=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for _ in range(1000):\n",
    "    A, B = random_contractive_system(n, d, rng)\n",
    "    assert operator_norm(A) < 1\n",
    "    K = np.zeros((d, n))\n",
    "    S = np.eye(n)\n",
    "    R = np.eye(d)\n",
    "    K_next = policy_iteration(A, B, K, S, R)\n",
    "    A_clp_next = A + B @ K_next\n",
    "    assert spectral_radius(A_clp_next) < 1\n",
    "    if operator_norm(A_clp_next) > 1:\n",
    "        print(\"counter example found\")\n",
    "        print(\"spectral_radius\", spectral_radius(A_clp_next), \"operator_norm\", operator_norm(A_clp_next))\n",
    "        break"
   ]
  }
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