Dynamic Optimization
Dynamic optimization solves optimization problems constrained by differential or differential-algebraic equations, such as parameter estimation, optimal control and model predictive control. In the simultaneous (direct transcription) approach the DAE is fully discretized, often by orthogonal collocation on finite elements, and the resulting large sparse nonlinear program is solved with an NLP solver such as an interior point method; the sequential approach instead integrates the model for each parameter guess and optimizes over the parameters alone.
Periodic boundary value problems for cyclic steady states (2024)
Tedesco and coworkers used a full-discretization, equation-oriented formulation (Pyomo.DAE with implicit Euler, solved by IPOPT) to find limit cycles of a periodically forced catalytic reactor, imposing equality of every state at the start and end of the period instead of integrating to steady state (Tedesco et al. 2024). Run times stayed nearly constant across forcing frequencies, while the sequential approach slowed sharply as frequency rose. Waveform parameters were optimized by an outer derivative-free (Nelder-Mead, Powell) or L-BFGS-B loop, and optimizing continuous waveforms directly within IPOPT gave the same optima. Summary: tedesco2024.
Simultaneous approach for neural DAE training (2026)
Lueg and coworkers apply Radau collocation and IPOPT to the training of hybrid models in which a neural network supplies unknown algebraic terms (Lueg et al. 2026). The network weights enter as dense, highly nonconvex constraints, which they handle with an initialization problem, external gray-box models and L-BFGS Hessian approximations; consistent initial conditions for an index-2 DAE are obtained within the NLP. They also show that the same simultaneous formulation, with slacks on the network output, can be used for inference to keep trajectories within physical bounds. Summary: lueg2026.