  Unambiguously distinguishing between nonorthogonal but linearly independent
quantum states is a challenging problem in quantum information processing. In
this work, an exact analytic solution to an optimum measurement problem
involving an arbitrary number of pure linearly independent quantum states is
presented. To this end, the relevant semi-definite programming task is reduced
to a linear programming one with a feasible region of polygon type which can be
solved via simplex method. The strength of the method is illustrated through
some explicit examples. Also using the close connection between the
Lewenstein-Sanpera decomposition(LSD) and semi-definite programming approach,
the optimal positive operator valued measure for some of the well-known
examples is obtain via Lewenstein-Sanpera decomposition method. {\bf Keywords:}
Optimal Unambiguous State Discrimination, Linear Programming,
Lewenstein-Sanpera decomposition. {\bf PACs Index: 03.67.Hk, 03.65.Ta, 42.50.-p
