Solution of linear equations and inequalities in idempotent vector spaces

Linear vector equations and inequalities are considered defined in terms of idempotent mathematics. To solve the equations, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces. The approach reduces the solution of the equation to that of an optimization problem in the idempotent algebra setting. Based on the approach, existence and uniqueness conditions are established for the solution of equations, and a general solution to both linear equations and inequalities are given. Finally, a problem of simultaneous solution of equations and inequalities is also considered.