Many variational problems in science and engineering are governed by constraints that encode essential physics. Equality constraints, such as incompressibility, and conservation of energy or momentum, underpin structure-preserving discretizations. Inequality constraints are equally pervasive, arising in free boundary problems, ice-sheet flow, groundwater filtration, and mathematical finance, as well as in regularization-based shock capturing for hyperbolic conservation laws. A unifying perspective is the geometric treatment of the structure governing the underlying physics, spanning conservation laws, exact sequences, and the nonsmoothness of contact, obstacles, and shocks. Designing high-order schemes that preserve structure, or resolve sharp features, with mesh-independent convergence remains an active research area. This minisymposium brings together researchers at the forefront of algorithmic and theoretical developments in physics- and structure-preserving numerical methods for variational problems, with an emphasis on: (a) compatible discretizations, optimization algorithms to achieve mesh independence, structure and constraint preservation, and sharp-feature resolution; (b) a priori and a posteriori error estimates; and (c) applications in contact mechanics, geophysics, electromagnetics, topology optimization, compressible fluid dynamics, and other areas. The session aims to identify open problems including adaptive high-order methods, efficient solvers for large-scale constrained problems, and the coupling of constrained and regularized variational formulations with other physical models.