Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equivalent conditions, and also propose some computationally cheap sufficient\,/\,necessary conditions. Our results extend the classical results on positive (semi-)definiteness of interval matrices. They may be useful for checking convexity or non-convexity in global optimization methods based on branch and bound framework and using interval techniques

System Identification and Sampling Strategies Applied to the Development of Compartment Models

Electrical Engineerin

A modification of the [alpha]BB method for box-constrained optimization and an application to inverse kinematics

For many practical applications it is important to determine not only a numerical approximation of one but a representation of the whole set of globally optimal solutions of a non-convex optimization problem. Then one element of this representation may be chosen based on additional information which cannot be formulated as a mathematical function or within a hierarchical problem formulation. We present such an application in the field of robotic design. This application problem can be modeled as a smooth box-constrained optimization problem. For determining a representation of the global optimal solution set with a predefined quality we modify the well known BB method. We illustrate the properties and give a proof for the finiteness and correctness of our modified BB method

SCEE 2008 book of abstracts : the 7th International Conference on Scientific Computing in Electrical Engineering (SCEE 2008), September 28 – October 3, 2008, Helsinki University of Technology, Espoo, Finland

This report contains abstracts of presentations given at the SCEE 2008 conference.reviewe

Spatial Interaction and Economic Growth

Spatial interaction is a central characteristic of economic activity. This thesis aims at providing new insights into the role of integration in the agglomeration-growth nexus and argues that integration is a multidimensional concept that pins down the impact of institutions to a spatial dimension. Also, the impact of genetic distance and data revisions on knowledge spillovers is discussed. Finally, knowledge spillovers are analyzed by applying a multi-region endogenous growth model to the US

Numerical Solution of Optimal Control Problems with Explicit and Implicit Switches

This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices

Exploring novel designs of NLP solvers: Architecture and Implementation of WORHP

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

Global Optimal Control Using Direct Multiple Shooting

The goal of this thesis is the development of a novel and efficient algorithm to determine the global optimum of an optimal control problem. In contrast to previous methods, the approach presented here is based on the direct multiple shooting method for discretizing the optimal control problem, which results in a significant increase of efficiency. To relax the discretized optimal control problems, the so-called alpha-branch-and-bound method in combination with validated integration is used. For the direct comparison of the direct single-shooting-based relaxations with the direct multipleshooting-based algorithm, several theoretical results are proven that build the basis for the efficiency increase of the novel method. A specialized branching strategy takes care that the additionally introduced variables due to the multiple shooting approach do not increase the size of the resulting branch-and-bound tree. An adaptive scaling technique of the commonly used Gershgorin method to estimate the eigenvalues of interval matrices leads to optimal relaxations and therefore leads to a general improvement of the alpha-branch-and-bound relaxations in a single shooting and a multiple shooting framework, as well as for the corresponding relaxations of non-dynamic nonlinear problems. To further improve the computational time, suggestions regarding the necessary second-order interval sensitivities are presented in this thesis, as well as a heuristic that relaxes on a subspace only. The novel algorithm, as well as the single-shooting-based alternative for a direct comparison, are implemented in a newly developed software package called GloOptCon. The new method is used to globally solve both a number of benchmark problems from the literature, and so far in the context of global optimal control still little considered applications. The additional problems pose new challenges either due to their size or due to having its origin in mixed integer optimal control with an integer-valued time-dependent control variable. The theoretically proven increase of efficiency is validated by the numerical results. Compared to the previous approach from the literature, the number of iterations for typical problems is more than halved, meanwhile the computation time is reduced by almost an order of magnitude. This in turn allows the global solution of significantly larger optimal control problems