ENH: Add LP solution method to DiscreteDP

quantecon/markov/_ddp_linprog_simplex.py

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+import numpy as np
+from numba import jit
+from .utilities import _find_indices
+from ..optimize.linprog_simplex import solve_tableau, PivOptions
+from ..optimize.pivoting import _pivoting
+
+
+@jit(nopython=True, cache=True)
+def ddp_linprog_simplex(R, Q, beta, a_indices, a_indptr, sigma,
+                        max_iter=10**6, piv_options=PivOptions(),
+                        tableau=None, basis=None, v=None):
+    r"""
+    Numba jit complied function to solve a discrete dynamic program via
+    linear programming, using `optimize.linprog_simplex` routines. The
+    problem has to be represented in state-action pair form with 1-dim
+    reward ndarray `R` of shape (n,), 2-dim transition probability
+    ndarray `Q` of shapce (L, n), and disount factor `beta`, where n is
+    the number of states and L is the number of feasible state-action
+    pairs.
+
+    The approach exploits the fact that the optimal value function is
+    the smallest function that satisfies :math:`v \geq T v`, where
+    :math:`T` is the Bellman operator, and hence it is a (unique)
+    solution to the linear program:
+
+    minimize::
+
+        \sum_{s \in S} v(s)
+
+    subject to ::
+
+        v(s) \geq r(s, a) + \beta \sum_{s' \in S} q(s'|s, a) v(s')
+                  \quad ((s, a) \in \mathit{SA}).
+
+    This function solves its dual problem:
+
+    maximize::
+
+        \sum_{(s, a) \in \mathit{SA}} r(s, a) y(s, a)
+
+    subject to::
+
+        \sum_{a: (s', a) \in \mathit{SA}} y(s', a) -
+            \sum_{(s, a) \in \mathit{SA}} \beta q(s'|s, a) y(s, a) = 1
+            \quad (s' \in S),
+
+        y(s, a) \geq 0 \quad ((s, a) \in \mathit{SA}),
+
+    where the optimal value function is obtained as an optimal dual
+    solution and an optimal policy as an optimal basis.
+
+    Parameters
+    ----------
+    R : ndarray(float, ndim=1)
+        Reward ndarray, of shape (n,).
+
+    Q : ndarray(float, ndim=2)
+        Transition probability ndarray, of shape (L, n).
+
+    beta : scalar(float)
+        Discount factor. Must be in [0, 1).
+
+    a_indices : ndarray(int, ndim=1)
+        Action index ndarray, of shape (L,).
+
+    a_indptr : ndarray(int, ndim=1)
+        Action index pointer ndarray, of shape (n+1,).
+
+    sigma : ndarray(int, ndim=1)
+        ndarray containing the initial feasible policy, of shape (n,).
+        To be modified in place to store the output optimal policy.
+
+    max_iter : int, optional(default=10**6)
+        Maximum number of iteration in the linear programming solver.
+
+    piv_options : PivOptions, optional
+        PivOptions namedtuple to set tolerance values used in the linear
+        programming solver.
+
+    tableau : ndarray(float, ndim=2), optional
+        Temporary ndarray of shape (n+1, L+n+1) to store the tableau.
+        Modified in place.
+
+    basis : ndarray(int, ndim=1), optional
+        Temporary ndarray of shape (n,) to store the basic variables.
+        Modified in place.
+
+    v : ndarray(float, ndim=1), optional
+        Output ndarray of shape (n,) to store the optimal value
+        function. Modified in place.
+
+    Returns
+    -------
+    success : bool
+        True if the algorithm succeeded in finding an optimal solution.
+
+    num_iter : int
+        The number of iterations performed.
+
+    v : ndarray(float, ndim=1)
+        Optimal value function (view to `v` if supplied).
+
+    sigma : ndarray(int, ndim=1)
+        Optimal policy (view to `sigma`).
+
+    """
+    L, n = Q.shape
+
+    if tableau is None:
+        tableau = np.empty((n+1, L+n+1))
+    if basis is None:
+        basis = np.empty(n, dtype=np.int_)
+    if v is None:
+        v = np.empty(n)
+
+    _initialize_tableau(R, Q, beta, a_indptr, tableau)
+    _find_indices(a_indices, a_indptr, sigma, out=basis)
+
+    # Phase 1
+    for i in range(n):
+        _pivoting(tableau, basis[i], i)
+
+    # Phase 2
+    success, status, num_iter = \
+        solve_tableau(tableau, basis, max_iter-n, skip_aux=True,
+                      piv_options=piv_options)
+
+    # Obtain solution
+    for i in range(n):
+        v[i] = tableau[-1, L+i] * (-1)
+
+    for i in range(n):
+        sigma[i] = a_indices[basis[i]]
+
+    return success, num_iter+n, v, sigma
+
+
+@jit(nopython=True, cache=True)
+def _initialize_tableau(R, Q, beta, a_indptr, tableau):
+    """
+    Initialize the `tableau` array.
+
+    Parameters
+    ----------
+    R : ndarray(float, ndim=1)
+        Reward ndarray, of shape (n,).
+
+    Q : ndarray(float, ndim=2)
+        Transition probability ndarray, of shape (L, n).
+
+    beta : scalar(float)
+        Discount factor. Must be in [0, 1).
+
+    a_indptr : ndarray(int, ndim=1)
+        Action index pointer ndarray, of shape (n+1,).
+
+    tableau : ndarray(float, ndim=2)
+        Empty ndarray of shape (n+1, L+n+1) to store the tableau.
+        Modified in place.
+
+    Returns
+    -------
+    tableau : ndarray(float, ndim=2)
+        View to `tableau`.
+
+    """
+    L, n = Q.shape
+
+    for j in range(L):
+        for i in range(n):
+            tableau[i, j] = Q[j, i] * (-beta)
+
+    for i in range(n):
+        for j in range(a_indptr[i], a_indptr[i+1]):
+            tableau[i, j] += 1
+
+    tableau[:n, L:-1] = 0
+
+    for i in range(n):
+        tableau[i, L+i] = 1
+        tableau[i, -1] = 1
+
+    for j in range(L):
+        tableau[-1, j] = R[j]
+
+    tableau[-1, L:] = 0
+
+    return tableau