  We consider the bail-out optimal dividend problem under fixed transaction
costs for a L\'evy risk model. Furthermore, we consider the version with a
constraint expected net present value of injected capital. To characterize the
solution to the aforementioned models, we first solve the bail-out optimal
dividend problem under transaction costs and capital injection and show the
optimality of reflected (c1; c2)- policies. Next, we introduce the dual
Lagrangian problem and show that the complementary slackness conditions are
satisfied, characterizing the optimal Lagrange multiplier. Finally, we
illustrate our findings with a series of numerical examples.
