  In this article we show that the new BPS equations for vortices, with nonzero
diagonal components of the stress tensor, obtained in \cite{Atmaja:2015lia} for
the generalized Maxwell-Higgs model can also be derived using the BPS
Lagrangian method developed in \cite{Atmaja:2015umo}. We add into the original
BPS Lagrangian $L_{BPS}=\int dQ$, which is a total derivative term, two
additional terms that are proportional to square of the first-derivative of
scalar effective field, $f'(r)^2$, and to a function that depends only on the
scalar effective field. These additional terms produce additional constraint
equations coming from Euler-Lagrange equations of the BPS Lagrangian. We apply
this procedure for the generalized Born-Infeld-Higgs model and show that the
total static energy, for the corresponding BPS equations, is finite if the
scalar potential $V< 2b^2$, with $b$ is the Born-Infeld parameter. We also
compute the energy-momentum tensor and show that its diagonal spatial
components in radial and angular directions are nonzero. Furthermore we show
that the conservation of energy-momentum does not produce new constraint
equation. We do the numerical analysis and found that for a large class of
solutions the scalar and gauge effective fields, $f(r)$ and $a(r)$, behave
nicely near the origin, but unfortunately they are infinite near the boundary.
We suggest that incorporate gravity into the action might resolve this problem
and other resolution is by considering BPS vortex in higher dimensional models.
We also suggest that the BPS Lagrangian method could be used to find BPS
equations for other solitons with nonzero stress tensor.
