We present a new "lifting" approach for the solution of nonlinear optimization problems (NLPs) that have objective and constraint functions with intermediate variables. Introducing these as additional degrees of freedom into the original problem, combined with adding suitable new constraints to ensure equivalence of the problems, we propose to solve this augmented system instead of the original system by a Newton-type method. This often offers advantages in terms of convergence rates and region of attraction. The main contribution of this article is an efficient algorithmic trick to generate the quantities needed for a Newton-type method on the augmented ("lifted") system with (a) almost no additional computational cost per iteration compared to a nonlifted Newton method, and (b) with negligible programming burden. We derive lifted schemes for Newton's method, as well as for constrained Gauss-Newton and adjoint based sequential quadratic programming (SQP) methods, and show equivalence of the new efficiently lifted approaches with "full-space" lifted Newton iterations. We establish conditions on the intermediate functions that imply faster local quadratic convergence for lifted versus nonlifted Newton iterations, a phenomenon often observed in practice but not yet explained theoretically. Finally, we compare numerically the behavior of the lifted approach with the nonlifted approach on several test problems, including a large scale example with 27 million intermediate variables. The algorithms and examples are available as open-source code in the C++ package LiftOpt.

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The application of optimization-based control methods such as nonlinear model predictive control (NMPC) to real-world process models is still a major computational challenge. In this paper, we present a new numerical optimization scheme suited for NMPC. The SQP-type approach uses an inexact constraint Jacobian in its iterations and is based on adjoint derivatives, that can be computed very efficiently. In comparison to a similar real-time algorithm based on directional sensitivities and an exact constraint Jacobian, the computational complexity is significantly reduced. Both algorithms are applied to the model of a thermally coupled distillation column for disturbance rejection. The results provide a proof-of-principle for the proposed adjoint-based optimization approach.

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