Inexact Newton-Type Optimization with Iterated Sensitivities

This paper presents and analyzes an Inexact Newton-type optimization method based on Iterated Sensitivities (INIS). A particular class of Nonlinear Programming (NLP) problems is considered, where a subset of the variables is defined by nonlinear equality constraints. The proposed algorithm considers any problem-specific approximation for the Jacobian of these constraints. Unlike other inexact Newton methods, the INIS-type optimization algorithm is shown to preserve the local convergence properties and the asymptotic contraction rate of the Newton-type scheme for the feasibility problem, yielded by the same Jacobian approximation. The INIS approach results in a computational cost which can be made close to that of the standard inexact Newton implementation. In addition, an adjoint-free (AF-INIS) variant of the approach is presented which, under certain conditions, becomes considerably easier to implement than the adjoint based scheme. The applicability of these results is motivated, specifically for dynamic optimization problems. In addition, the numerical performance of a specific open-source implementation is illustrated

Discrete Adjoints: Theoretical Analysis, Efficient Computation, and Applications

The technique of automatic differentiation provides directional derivatives and discrete adjoints with working accuracy. A complete complexity analysis of the basic modes of automatic differentiation is available. Therefore, the research activities are focused now on different aspects of the derivative calculation, as for example the efficient implementation by exploitation of structural information, studies of the theoretical properties of the provided derivatives in the context of optimization problems, and the development and analysis of new mathematical algorithms based on discrete adjoint information. According to this motivation, this habilitation presents an analysis of different checkpointing strategies to reduce the memory requirement of the discrete adjoint computation. Additionally, a new algorithm for computing sparse Hessian matrices is presented including a complexity analysis and a report on practical experiments. Hence, the first two contributions of this thesis are dedicated to an efficient computation of discrete adjoints. The analysis of discrete adjoints with respect to their theoretical properties is another important research topic. The third and fourth contribution of this thesis focus on the relation of discrete adjoint information and continuous adjoint information for optimal control problems. Here, differences resulting from different discretization strategies as well as convergence properties of the discrete adjoints are analyzed comprehensively. In the fifth contribution, checkpointing approaches that are successfully applied for the computation of discrete adjoints, are adapted such that they can be used also for the computation of continuous adjoints. Additionally, the fifth contributions presents a new proof of optimality for the binomial checkpointing that is based on new theoretical results. Discrete adjoint information can be applied for example for the approximation of dense Jacobian matrices. The development and analysis of new mathematical algorithms based on these approximate Jacobians is the topic of the sixth contribution. Is was possible to show global convergence to first-order critical points for a whole class of trust-region methods. Here, the usage of inexact Jacobian matrices allows a considerable reduction of the computational complexity

Approximation methodologies for explicit model predictive control of complex systems

This thesis concerns the development of complexity reduction methodologies for the application of multi-parametric/explicit model predictive (mp-MPC) control to complex high fidelity models. The main advantage of mp-MPC is the offline relocation of the optimization task and the associated computational expense through the use of multi-parametric programming. This allows for the application of MPC to fast sampling systems or systems for which it is not possible to perform online optimization due to cycle time requirements. The application of mp-MPC to complex nonlinear systems is of critical importance and is the subject of the thesis. The first part is concerned with the adaptation and development of model order reduction (MOR) techniques for application in combination to mp-MPC algorithms. This first part includes the mp-MPC oriented use of existing MOR techniques as well as the development of new ones. The use of MOR for multi-parametric moving horizon estimation is also investigated. The second part of the thesis introduces a framework for the ‘equation free’ surrogate-model based design of explicit controllers as a possible alternative to multi-parametric based methods. The methodology relies upon the use of advanced data-classification approaches and surrogate modelling techniques, and is illustrated with different numerical examples.Open Acces