# Solves MDP with average reward using relative value iteration algorithm

### Description

Solves MDP with average reward using relative value iteration algorithm

### Usage

1 | ```
mdp_relative_value_iteration(P, R, 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. |

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

`max_iter` |
(optional) : maximum number of iterations. max_iter is an integer greater than 0. By default, max_iter is set to 1000. |

### Details

mdp_relative_value_iteration applies the relative value iteration algorithm to solve MDP with average reward. The algorithm consists in solving optimality equations iteratively. Iterating is stopped when an epsilon-optimal policy is found or after a specified number (max_iter) of iterations is done.

### Value

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

`average_reward ` |
average reward of the optimal policy. average_reward is a real. |

`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_relative_value_iteration(P, R)
# 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_relative_value_iteration(P, R)
``` |