A condensing approach for linear-quadratic optimization with geometric constraints

Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and cardinality constraints, among others. In particular, we cover also situations where parts of the constraints are nonconvex and possibly complicated, but it is practical to compute projections onto this nonconvex set. Our approach combines the augmented Lagrangian framework with a solver-agnostic structure-exploiting subproblem reformulation. While convergence guarantees follow from the former, the proposed condensing technique leads to significant improvements in computational performance.