VALUE-ITERATION ALGORITHM

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Theorem 663 In the standard value-iteration algorithm the lower and upper bounds satisfy mk+1 mk and Mk+1 Mk for all k 1 Proof By the de nition of policy R(n), Vn (i) = ci (Ri (n)) +

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pij (Ri (n))Vn 1 (j ),

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(668)

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In the same way as (665) was obtained, we nd for any policy R that ci (Ri ) +

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pij (Ri )Vn 1 (j ) Vn (i),

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(669)

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Taking n = k in (668) and taking n = k + 1 and R = R(k) in (669) gives Vk+1 (i) Vk (i)

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pij (Ri (k)){Vk (j ) Vk 1 (j )},

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(6610)

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Similarly, by taking n = k + 1 in (668) and taking n = k and R = R(k + 1) in (669), we nd Vk+1 (i) Vk (i)

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pij (Ri (k + 1)) {Vk (j ) Vk 1 (j )} ,

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(6611)

Since Vk (j ) Vk 1 (j ) Mk for all j I and j I pij (Ri (k)) = 1, it follows from (6610) that Vk+1 (i) Vk (i) Mk for all i I This gives Mk+1 Mk Similarly, we obtain from (6611) that mk+1 mk Data transformation The periodicity issue can be circumvented by a perturbation of the one-step transition probabilities The perturbation technique is based on the following two observations First, a recurrent state allowing for a direct transition to itself must be aperiodic Second, the relative frequencies at which the states of a Markov chain are visited do not change when the state changes are delayed with a constant factor and the probability of a self-transition is accordingly enlarged In other words, if the one-step transition probabilities pij of a Markov chain {Xn } are perturbed as pij = pij for j = i and pii = pii + 1 for some constant with 0 < < 1, the perturbed Markov chain {X n } with one-step transition probabilities pij is aperiodic and has the same equilibrium probabilities as the original Markov chain {Xn } (verify) Thus a Markov decision model involving periodicities may be perturbed as follows Choosing some constant with 0 < < 1, the state space, the action sets, the one-step costs and the one-step transition probabilities of the perturbed

DISCRETE-TIME MARKOV DECISION PROCESSES

Markov decision model are de ned by I = I, A(i) = A(i), ci (a) = ci (a), pij (a) = i I, a A(i) and i I , j = i, a A(i) and i I , j = i, a A(i) and i I

pij (a), pij (a) + 1 ,

For each stationary policy, the associated Markov chain {X n } in the perturbed model is aperiodic It is not dif cult to verify that for each stationary policy the average cost per time unit in the perturbed model is the same as that in the original model For the unichain case this is an immediate consequence of the representation (627) for the average cost and the fact that for each stationary policy the Markov chain {X n } has the same equilibrium probabilities as the Markov chain {Xn } in the original model For the multichain case, a similar argument can be used to show that the two models are in fact equivalent Thus the value-iteration algorithm can be applied to the perturbed model in order to solve the original model In speci c problems involving periodicities, the optimal value of is usually not clear beforehand; empirical investigations indicate that = 1 is usually a satisfactory choice 2 Modi ed value iteration with a dynamic relaxation factor Value iteration does not have the fast convergence of policy iteration The number of iterations required by the value-iteration algorithm is problem dependent and increases when the number of problem states gets larger Also, the tolerance number in the stopping criterion affects the number of iterations required The stopping criterion should be based on the lower and upper bounds mn and Mn but not on any repetitive behaviour of the generated policies R(n) The convergence rate of value iteration can often be accelerated by using a relaxation factor, such as in successive overrelaxation for solving a single system of linear equations Then at the nth iteration a new approximation to the value function Vn (i) is obtained by using both the previous values Vn 1 (i) and the residuals Vn (i) Vn 1 (i) It is possible to select dynamically a relaxation factor and thus avoid the experimental determination of the best value of a xed relaxation factor The following modi cation of the standard value-iteration algorithm can be formulated Steps 0, 1, 2 and 3 are as before, while step 4 of the standard value-iteration algorithm is modi ed as follows Step 4(a) Determine the states u and v such that Vn (u) Vn 1 (u) = mn and compute the relaxation factor = Mn mn + Mn mn , j I {puj (Ru ) pvj (Rv )}{Vn (j ) Vn 1 (j )} and Vn (v) Vn 1 (v) = Mn