The influence of cyclic loading and glass stability on structural relaxation and yielding transition in amorphous alloys was investigated using molecular dynamics simulations. We considered a binary mixture cooled deep into the glass phase and subjected to cyclic shear deformation where strain varies periodically but remains positive. We found that rapidly cooled glasses under asymmetric cyclic shear gradually evolve towards states with lower potential energy and finite stress at zero strain. At the strain amplitude just below a critical value, the rescaled distributions of nonaffine displacements converge to a power-law decay with an exponent of about -2 upon increasing number of cycles. By contrast, more stable glasses yield at lower stra...