  Relaxation Runge-Kutta methods reproduce a fully discrete dissipation (or
conservation) of entropy for entropy stable semi-discretizations of nonlinear
conservation laws. In this paper, we derive the discrete adjoint of relaxation
Runge-Kutta schemes, which are applicable to discretize-then-optimize
approaches for optimal control problems. Furthermore, we prove that the derived
discrete relaxation Runge-Kutta adjoint preserves time-symmetry when applied to
linear skew-symmetric systems of ODEs. Numerical experiments verify these
theoretical results while demonstrating the importance of appropriately
treating the relaxation parameter when computing the discrete adjoint.
