  In the fourth-generation storage ring light sources (4GLSs), associated with
the extremely strong nonlinearities inherent in the multi-bend achromat design,
the dynamic acceptance is usually small and it is difficult to implement
traditional off-axis local-bump injection. To release the requirement on
dynamic acceptance, on-axis longitudinal injection schemes have been explored.
In this paper, we present a multi-objective optimization of longitudinal
injection with a multi-frequency RF system, based on the parameters of the
Southern Advanced Photon Source, a 4GLS proposed in China. We show that by
treating the optimal bunch lengthening condition as an optimizing objective
rather than a condition that must be satisfied, a plethora of feasible
candidate solutions can be found, showing different trade-offs among
multi-objectives. From these candidate solutions, one can find an optimal RF
parameter setting that is most adapted to a specific 4GLS physics design and
the available technical level of injection kicker. Especially, it would be
feasible to realize longitudinal injection within a static bucket enabled by a
double-frequency RF system, suggesting an attractive longitudinal injection
option for 4GLSs.
