  It is shown that for a given bipartite density matrix and by choosing a
suitable separable set (instead of product set) on the separable-entangled
boundary, optimal Lewenstein-Sanpera (L-S) decomposition can be obtained via
optimization for a generic entangled density matrix. Based on this, We obtain
optimal L-S decomposition for some bipartite systems such as $2\otimes 2$ and
$2\otimes 3$ Bell decomposable states, generic two qubit state in Wootters
basis, iso-concurrence decomposable states, states obtained from BD states via
one parameter and three parameters local operations and classical
communications (LOCC), $d\otimes d$ Werner and isotropic states, and a one
parameter $3\otimes 3$ state. We also obtain the optimal decomposition for
multi partite isotropic state. It is shown that in all $2\otimes 2$ systems
considered here the average concurrence of the decomposition is equal to the
concurrence. We also show that for some $2\otimes 3$ Bell decomposable states
the average concurrence of the decomposition is equal to the lower bound of the
concurrence of state presented recently in [Buchleitner et al,
quant-ph/0302144], so an exact expression for concurrence of these states is
obtained. It is also shown that for $d\otimes d$ isotropic state where
decomposition leads to a separable and an entangled pure state, the average
I-concurrence of the decomposition is equal to the I-concurrence of the state.
  Keywords: Quantum entanglement, Optimal Lewenstein-Sanpera decomposition,
Concurrence, Bell decomposable states, LOCC}
  PACS Index: 03.65.Ud

Quantum Physics