# Solves discounted MDP using modified policy iteration algorithm

### Description

Solves discounted MDP using modified policy iteration algorithm

### Usage

1 | ```
mdp_policy_iteration_modified(P, R, discount, epsilon, max_iter)
``` |

### Arguments

`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 to be done. max_iter is an integer greater than 0. By default, max_iter = 1000. |

### Details

mdp_policy_iteration_modified applies the modified policy iteration algorithm to solve discounted MDP. The algorithm consists, like policy iteration one, in improving the policy iteratively but in policy evaluation few iterations (max_iter) of value function updates done. Iterating is stopped when an epsilon-optimal policy is found.

### Value

`V ` |
optimal value fonction. V is a S length vector |

`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 iterations |

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

### Examples

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_policy_iteration_modified(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_policy_iteration_modified(P, R, 0.9)
``` |

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