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

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