Arbitrarily tight aBB underestimators of general non-linear functions over sub-optimal domains

In this paper we explore the construction of arbitrarily tight αBB relaxations of C2 general non-linear non-convex functions. We illustrate the theoretical challenges of building such relaxations by deriving conditions under which it is possible for an αBB underestimator to provide exact bounds. We subsequently propose a methodology to build αBB underestimators which may be arbitrarily tight (i.e., the maximum separation distance between the original function and its underestimator is arbitrarily close to 0) in some domains that do not include the global solution (defined in the text as “sub-optimal”), assuming exact eigenvalue calculations are possible. This is achieved using a transformation of the original function into a μ-subenergy function and the derivation of αBB underestimators for the new function. We prove that this transformation results in a number of desirable bounding properties in certain domains. These theoretical results are validated in computational test cases where approximations of the tightest possible μ-subenergy underestimators, derived using sampling, are compared to similarly derived approximations of the tightest possible classical αBB underestimators. Our tests show that μ-subenergy underestimators produce much tighter bounds, and succeed in fathoming nodes which are impossible to fathom using classical αBB

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