Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York

Analyse und Restrukturierung eines Verfahrens zur direkten Loesung von Optimal-Steuerungsproblemen (the theory of MUSCOD in a nutshell)

MUSCOD (MUltiple Shooting COde for Direct Optimal Control) is the implementation of an algorithm for the direct solution of optimal control problems. The method is based on multiple shooting combined with a sequential quadratic programming (SQP) technique; its original version was developed in the early 1980s by Plitt under the supervision of Bock. The following report is intended to describe the basic aspects of the underlying theory in a concise but readable form. Such a description is not yet available: The paper by Bock and Plitt gives a good overview of the method, but it leaves out too many important details to be a complete reference, while the diploma thesis by Plitt, on the other hand, presents a fairly complete description, but is rather difficult to read. Throughout the present document, emphasis is given to a clear presentation of the concepts upon which MUSCOD is based. An effort has been made to properly reflect the structure of the algorithm and to give a good motivation for its fundamental parts. Of course, recent developments have been taken into account, and it should be noted that some of the material presented here has not been previously published. Proofs which are readily found elsewhere have not been included. It is hoped that this tutorial description will help new users of MUSCOD to quickly become familiar with the necessary theory, and perhaps it will also be useful for people who want to modify or extend the existing algorithm. (orig.)Available from TIB Hannover: RR 1606(96-19) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Fast direct methods for real-time optimization of chemical processes

Some specialized direct methods for real-time optimization of dynamic chemical processes are presented. The methods are based on a boundary value problem approach to the solution of large, multistage optimal control and design optimization problems for processes described by DAE models (Leineweber et al., 1997). This simultaneous solution strategy uses a piecewise parametrization of the control functions and a multiple shooting discretization of the DAEs, combined with a specifically tailored SQP technique (Plitt, 1981; Bock and Plitt, 1984; Bock et al., 1988; Leineweber, 1995). The existing general approach has now been adapted to the specific requirements of real-time optimization through the development of suitable strategies to minimize the on-line computational effort. In a typical real-time context, e.g., an on-line reoptimization after disturbances, the resulting new methods are much faster than off-line optimization strategies, because they exploit precalculated information about the undisturbed nominal solution in order to greatly accelerate the solution process. This kind of previous information is not available in the off-line case. The new on-line strategies have been implemented within the modular optimal control package MUSCOD-II (Leineweber, 1995; Leineweber et al., 1997). A comparative study of these techniques has been done for the reoptimization of a fed-batch fermentation process. The results indicate that in particular the use of precalculated exact Hessians in so-called HOT-starts has great potential for the on-line optimization of complex chemical processes - here the total computational cost of the reoptimization was reduced by almost an order of magnitude compared to the off-line strategy. Clearly, such an acceleration may be of crucial importance in time-critical on-line applications. (orig.)Available from TIB Hannover: RR 1606(97-21) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

LIBLAC - structured data types and basic operations for numerical linear algebra in an ANSI C/Fortran 77 environment

The C/Fortran library LIBLAC provides a set of convenient, structured data types and basic operations for numerical linear algebra. The library is intended to facilitate the ANSI C implementation of high-level applications which involve hierarchically structured data like partitioned vectors or block matrices. By maintaining compatibility with Fortran 77 conventions (e.g., matrices are stored by columns, and vectors may have arbitrary memory strides), the task of sharing data with existing Fortran 77 code is greatly simplified. Basic operations for data access, data copying/conversion, and input/output are supported in fully generic form. In addition, safe and user-firendly interfaces to Level 1-2 BLAS [7, 3, 4] and LAPACK [1] are provided, i.e., the use of existing libraries of highly efficient Fortran 77 subroutines is encouraged for performing common lower level computations. LIBLAC is easily portable to platforms supporting ANSI C and Standard Fortran 77, and it fully encapsulates the platform-dependent details of the C/Fotran interface, thereby resolving the inherent portability problems of mixed-language programs. The performance of typical applications is competitive with Fortran 77/BLAS. (orig.)Available from TIB Hannover: RR 1606(96-56) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Pt. 1: Theoretical aspects

SIGLEAvailable from TIB Hannover: RR 1606(2001,23) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Numerical Solution of Optimal Control Problems with Constant Control Delays

Delayed differential equations, Delayed optimal control, Numerical optimization, C63, C61,

Efficient techniques for the optimization of complex chemical processes

Efficient and robust techniques for the optimization of complex chemical processes are presented. In particular, we address the solution of large, multistage optimal control problems for processes described by DAE models. Our boundary value problem approach (a simultaneous solution strategy) is based on a piecewise representation of the control functions and a multiple shooting discretization of the DAEs, combined with a specifically tailored SQP technique. The inherent problem structure is exploited on various levels, leading to an extremely efficient overall method. A reliable and easy-to-use implementation of this strategy is provided by the recently developed modular optimal control package MUSCOD-II. Two real-life applications of MUSCOD-II are discussed: 1. Optimal control of a fed-batch fermentation for the production of Nikkomycin. 2. Optimization of a hydrogen-air combustion process in a 1D steady-state tube reactor. The first application provides a realistic bioprocess optimization problem, and our method is shown to be so fast that it could be even used for the online optimization of such processes. The second problem demonstrates the suitability of our approach for the stiff differential equation models that are common in chemical kinetics applications. (orig.)SIGLEAvailable from TIB Hannover: RR 1606(96-40) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman