In this talk, we present higher-order bounds preserving time-implicit discontinuous Galerkin (DG) discretizations for the nonlinear degenerate parabolic equations and the reactive Euler equations. Using Lagrange multipliers the conditions imposed by the positivity preserving limiters are directly coupled to a DG discretization combined with a Diagonally Implicit Runge-Kutta time integration method. The positivity preserving DG discretization is then reformulated as a Karush-Kuhn-Tucker (KKT) problem, which is frequently encountered in constrained optimization. Since the limiter is only active in areas where positivity must be enforced it does not affect the higher-order DG discretization elsewhere. The resulting non-smooth nonlinear algebraic equations have, however, a different structure compared to most constrained optimization problems. We develop an efficient active set semi-smooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Convergence of this semi-smooth Newton method is proven using a specially designed quasi-directional derivative of the time-implicit positivity preserving DG discretization. Numerical results are shown to demonstrate that the bounds preserving DIRK-DG discretizations are higher order accurate for smooth solutions and also efficient for stiff problems with discontinuities.