Exploiting Multistage Optimization Structure in Proximal Solvers

This paper presents an efficient structure-exploiting algorithm for multistage optimization problems. The proposed method extends existing approaches by supporting full coupling between stages and global decision variables in the cost, as well as equality and inequality constraints. The algorithm is implemented as a new backend in the PIQP solver and leverages a specialized block-tri-diagonal-arrow Cholesky factorization within a proximal interior-point framework to handle the underlying problem structure efficiently. The implementation features automatic structure detection and seamless integration with existing interfaces. Numerical experiments demonstrate significant performance improvements, achieving up to 13x speed-up compared to a generic sparse backend and matching/exceeding the performance of the state-of-the-art specialized solver HPIPM. The solver is particularly effective for applications such as model predictive control, robust scenario optimization, and periodic optimization problems.