  We show that surface solitons in the one-dimensional nonlinear Schr\"odinger
equation with truncated complex periodic potential can be stabilized by linear
homogeneous losses, which are necessary to balance gain in the near-surface
channel arising from the imaginary part of potential. Such solitons become
stable attractors when the strength of homogeneous losses acquires values from
a limited interval and they exist in focusing and defocusing media. The domains
of stability of surface solitons shrink with increase of the amplitude of
imaginary part of complex potential.
