Parallel cyclic reduction decomposition for dynamic optimization problems

Abstract

Direct transcription of dynamic optimization problems, with differential-algebraic equations discretized and written as algebraic constraints, can create very large nonlinear optimization problems. When this discretized optimization problem is solved with an NLP solver, such as IPOPT, the dominant computational cost often lies in solving the linear system that generates Newton steps for the KKT system. Computational cost and memory constraints for this linear system solution raise many challenges as the system size increases. On the other hand, the linear KKT system for our dynamic optimization problem is sparse and structured, and can be permuted to form a block tridiagonal matrix. This study explores a parallel decomposition strategy for block tridiagonal systems that is based on cyclic reduction (CR) factorization of the KKT matrix. The classical CR method has good observed performance, but its numerical stability properties need further study for our KKT system. Finally, we discuss modifications to the CR decomposition that improve performance, and we apply the approach to four industrially relevant case studies. On the largest problem, a parallel speedup of a factor of four is observed when using eight processors.