Gerdts, Matthias
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Gerdts, Matthias
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Gerdts, Matthias
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Item-typ:Veröffentlichung, Korrekturverfahren zur numerischen Lösung nichtlinearer Optimierungsprobleme mittels Methoden der parametrischen Sensitivitätsanalyse(2012-02-24); ; ; The nonlinear optimization has becoming of the key technologies of the field of applied mathematics. There are various numbers of applications in different fields of science, e.g. economics, medicine and engineering. Also industry is using mathematical algorithms for solving nonlinear optimization problems. Especially in the aerospace industry nonlinear optimization is frequently used for optimizing trajectories. The aim of this thesis is to extend an existing method for solving nonlinear optimization problems by using parametric sensitivity analysis. At first the iterative SQP-method WORHP for solving nonlinear optimization problems is described and introduced. Then it is shown that the sensitivity derivatives of a quadratic subproblem within WORHP can be computed efficiently with almost no extra costs. The subproblem is now treated as a perturbed problem. In the thesis several different perturbations are introduced. With the sensitivity derivatives it is now possible to compute an approximation of the solution of the perturbed quadratic subproblem. This solution is used within the overall nonlinear optimization method for obtaining a new iterate. The numerical results show that it is possible to reduce of number of iterations and sometimes it also possible to reduce the time needed for solving a problem.Dissertation908 275 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Exploring novel designs of NLP solvers: Architecture and Implementation of WORHP(2013-04-25); ; ; Mathematical Optimization in general and Nonlinear Programming in particular, are applied by many scientific disciplines, such as the automotive sector, the aerospace industry, or the space agencies. With some established NLP solvers having been available for decades, and with the mathematical community being rather conservative in this respect, many of their programming standards are severely outdated. It is safe to assume that such usability shortcomings impede the wider use of NLP methods; a representative example is the use of static workspaces by legacy FORTRAN codes. This dissertation gives an account of the construction of the European NLP solver WORHP by using and combining software standards and techniques that have not previously been applied to mathematical software to this extent. Examples include automatic code generation, a consistent reverse communication architecture and the elimination of static workspaces. The result is a novel, industrial-grade NLP solver that overcomes many technical weaknesses of established NLP solvers and other mathematical software.Dissertation1029 419 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, On the concepts of quality for local solutions of nonlinear programs(2025-10-10); ; This thesis covers the topic of assessing the quality of local solutions in both standard and parametric nonlinear programming. From a theoretical perspective, a formal definition of a quality criterion for local solutions of nonlinear programs is introduced, and the notions of primary and secondary quality criteria are distinguished. Since the primary criterion is inherently associated with the optimality of local solutions, the main focus is on secondary criteria. In particular, over 25 existing secondary quality criteria are systematically reviewed and analyzed. These criteria, arising in both non-parametric and parametric nonlinear programming, are derived from the fields of dynamical systems, robust optimization, stochastic optimization, and parametric sensitivity analysis. From an applied perspective, two novel secondary quality criteria are proposed: the maximum radius of attraction (MRoA) and the parametric stability score (PSS). MRoA is a measure of quality for local solutions of non-parametric nonlinear programs. It is defined as the radius of the largest ball centered at a local solution such that a given optimization algorithm, once initialized inside that ball, is guaranteed never to escape it and is expected to converge to the same solution. PSS, in turn, quantifies the quality of local solutions of parametric nonlinear programs. It is defined as the maximum magnitude of a perturbation of the nominal parameter, such that for any smaller perturbation, the solutions and corresponding optimal costs of the nominal and perturbed problems remain within prescribed distances. The utility of the PSS concept is demonstrated using an optimal control problem for accomplishing the swing-up maneuver of the pendulum on a cart system. For both proposed criteria, connections to the field of online optimization are established, emphasizing their potential to support informed decision-making.Dissertation87 61 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Effiziente Ableitungsbestimmung bei hochdimensionaler nichtlinearer Optimierung(2011-06-17); ; ; Calculating or approximating derivative information in an efficient way is one of the major tasks when high dimensional nonlinear optimization problems are to be solved. The thesis shows various new ways both in the area of finite differences and in the area of update formalae to tackle this problem. One of the results in this thesis is the development of a sparse BFGS-similar method for which superlinear convergence properties are shown.Dissertation911 261 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Limited Memory BFGS method for Sparse and Large-Scale Nonlinear Optimization(2014-09-16); ; ; Optimization-based control systems are used in many areas of application, including aerospace engineering, economics, robotics and automotive engineering. This work was motivated by the demand for a large-scale sparse solver for this problem class. The sparsity property of the problem is used for the computational efficiency regarding performance and memory consumption. This includes an efficient storing of the occurring matrices and vectors and an appropriate approximation of the Hessian matrix, which is the main subject of this work. Thus, a so-called the limited memory BFGS method has been developed. The limited memory BFGS method, has been implemented in a software library for solving the nonlinear optimization problems, WORHP. Its solving performance has been tested on different optimal control problems and test sets.Dissertation662 213 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Online-Optimierung in der Mehrsystemdynamik unter einer erweiterten Klasse von Nebenbedingungen : Optimale Bahnplanung eines deckengeführten Bediensystems in der Hochregallagertechnik(2012-05-18); ; ; In order to facilitate a new concept for a high rack warehouses with ceiling rooted storage systems, reference trajectories have to be provided to ensure a controlled swing of the crane system and a steady state at the final point. These trajectories are obtained solving optimal control problems. Due to safety reasons alternative trajectories have to be calculated and have to be controlled to bring the crane system to an oscillation-free position from any state of the main trajectory. The aim of this thesis is to extend the constraints of the existing optimal control problem by using the classic description of linear quadratic regulatorproblems. Hence, the optimal closed-loop control and the closed-loop behavior are embedded in the global optimal control problem description. One main result in this thesis is the proof of the necessary continuous differentiability of this advanced formulation of constraints. In conclusion the feasibility with some coupling effects between the control law and the optimal trajectory are presented with respect to safety requirements and time of computation just as robustness of the system and the controller.Dissertation512 410