Abstract. We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. Additionally, however, we also consider invariant preservation over persistent, simultaneous, and/or frequent boundary interactions. Appropriately formulating geometric methods for these cases has long remained challenging due the inherent nonsmoothness and one-sided conditions that they impose. To resolve these issues we thus focus both on symplectic-momentum preserving behavior and the preservation of additional structures, unique to the inequality constrained setting. Toward these goals we introduce, for the first time, a fully nonsmooth, discrete Hamilton's principle and obtain an associated framework for composing geometric numerical integration methods for inequality-equality-constrained systems. Applying this framework, we formulate a new family of geometric numerical integration methods that, by construction, preserve momentum and equality constraints and are observed to retain good long-term energy behavior. Along with these standard geometric properties, the derived methods also enforce multiple simultaneous inequality constraints, obtain smooth unilateral motion along constraint boundaries, and allow for both nonsmooth and smooth boundary approach and exit trajectories. Numerical experiments are presented to illustrate the behavior of these methods on difficult test examples where both smooth and nonsmooth active constraint modes persist with high frequency.Key words. geometric integration, Hamiltonian systems, inequality constraints, nonsmooth systems, impact, contact dynamics AMS subject classifications. 65P10, 65L05, 82B80 DOI. 10.1137/100800105 1. Introduction. Recent attention has focused on geometric numerical integration methods that closely preserve invariants of continuous systems (Sanz-Serna and Calvo (1994);Hairer, Lubich, and Wanner (2002); Leimkuhler and Reich (2004)). In particular, symplectic-momentum preserving integration schemes have been successfully applied over a wide range of applications. In the unconstrained setting, these methods exactly preserve the symplectic form and momenta and maintain good long-term energetic behavior by approximately conserving energy (up to an additive constant) over long spans of simulation.While extensions of symplectic methods to equality-constrained systems have been extensively investigated (Hairer, Lubich, and Wanner (2002); Leimkuhler and Reich (2004)), appropriately formulating symplectic methods to include inequality constraints continues to remain a challenging problem. The difficulty in treating these latter cases stems from the inherent nonsmoothness imposed by such conditions. Stewart (2000), in particular, notes that, despite good conservative properties in the smooth setting, symplectic methods do not necessarily translate easily into nonsmooth problems. Direct extensions of existing symplectic methods (e....