  We show that the entanglement witnesses based on local orthogonal observables
which are introduced in [S. Yu and N.-L. Liu, Phys. Rev. Lett. 95, 150504
(2005)] and [O. G\"uhne, M. Mechler, G. T\'oth and P. Adam, Phys. Rev. A 74,
010301 (2006)] in linear and nonlinear forms can be optimized, respectively. As
applications, we calculate the optimal nonlinear witnesses of pure bipartite
states and show a lower bound on the I-concurrence of bipartite higher
dimensional systems with our method.
