def modified_policy_iteration(self, v_init=None, epsilon=None,
                                  max_iter=None, k=20):
        if self.beta == 1:
            raise NotImplementedError(self._error_msg_no_discounting)

        if max_iter is None:
            max_iter = self.max_iter
        if epsilon is None:
            epsilon = self.epsilon

        def span(z):
            return z.max() - z.min()

        def midrange(z):
            return (z.min() + z.max()) / 2

        v = np.empty(self.num_states)
        if v_init is None:
            v[:] = self.R[self.R > -np.inf].min() / (1 - self.beta)
        else:
            v[:] = v_init

        u = np.empty(self.num_states)
        sigma = np.empty(self.num_states, dtype=int)

        try:
            tol = epsilon * (1-self.beta) / self.beta
        except ZeroDivisionError:  # Raised if beta = 0
            tol = np.inf

        for i in range(max_iter):
            # Policy improvement
            self.bellman_operator(v, Tv=u, sigma=sigma)
            diff = u - v
            if span(diff) < tol:
                v[:] = u + midrange(diff) * self.beta / (1 - self.beta)
                break
            # Partial policy evaluation with k iterations
            self.operator_iteration(T=self.T_sigma(sigma), v=u, max_iter=k)
            v[:] = u

        num_iter = i + 1

        res = DPSolveResult(v=v,
                            sigma=sigma,
                            num_iter=num_iter,
                            mc=self.controlled_mc(sigma),
                            method='modified policy iteration',
                            epsilon=epsilon,
                            max_iter=max_iter,
                            k=k)

        return res
Solve the optimization problem by modified policy iteration. See
dataset/ETHPy150Open QuantEcon/QuantEcon.py/quantecon/markov/ddp.py/DiscreteDP.modified_policy_iteration