Solves discounted MDP with value iteration algorithm

1 | ```
mdp_value_iteration(P, R, discount, epsilon, max_iter, V0)
``` |

`P` |
transition probability array. P can be a 3 dimensions array [S,S,A] or a list [[A]], each element containing a sparse matrix [S,S]. |

`R` |
reward array. R can be a 3 dimensions array [S,S,A] or a list [[A]], each element containing a sparse matrix [S,S] or a 2 dimensional matrix [S,A] possibly sparse. |

`discount` |
discount factor. discount is a real number which belongs to [0; 1[. For discount equals to 1, a warning recalls to check conditions of convergence. |

`epsilon` |
(optional) : search for an epsilon-optimal policy. epsilon is a real in ]0; 1]. By default, epsilon = 0.01. |

`max_iter` |
(optional) : maximum number of iterations. max_iter is an integer greater than 0. If the value given in argument is greater than a computed bound, a warning informs that the computed bound will be considered. By default, if discount is not egal to 1, a bound for max_iter is computed, if not max_iter = 1000. |

`V0` |
(optional) : starting value function. V0 is a (Sx1) vector. By default, V0 is only composed of 0 elements. |

mdp_value_iteration applies the value iteration algorithm to solve discounted MDP. The algorithm consists in solving Bellman's equation iteratively. Iterating is stopped when an epsilon-optimal policy is found or after a specified number (max_iter) of iterations.

`policy ` |
optimal policy. policy is a S length vector. Each element is an integer corresponding to an action which maximizes the value function. |

`iter ` |
number of done iterations. |

`cpu_time ` |
CPU time used to run the program. |

1 2 3 4 5 6 7 8 9 10 11 12 | ```
# With a non-sparse matrix
P <- array(0, c(2,2,2))
P[,,1] <- matrix(c(0.5, 0.5, 0.8, 0.2), 2, 2, byrow=TRUE)
P[,,2] <- matrix(c(0, 1, 0.1, 0.9), 2, 2, byrow=TRUE)
R<- matrix(c(5, 10, -1, 2), 2, 2, byrow=TRUE)
mdp_value_iteration(P, R, 0.9)
# With a sparse matrix
P <- list()
P[[1]] <- Matrix(c(0.5, 0.5, 0.8, 0.2), 2, 2, byrow=TRUE, sparse=TRUE)
P[[2]] <- Matrix(c(0, 1, 0.1, 0.9), 2, 2, byrow=TRUE, sparse=TRUE)
mdp_value_iteration(P, R, 0.9)
``` |

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