(599a) Exploiting Structure in Direct Simultaneous Methods for Global Dynamic Optimization

To date, a majority of the research in deterministic global optimization for dynamic systems has focused on direct sequential methods, mainly direct single-shooting [1-4] and more recently direct multiple-shooting [5]. Such sequential methods rely on set-propagation techniques in order to enclose the reachable set of the embedded dynamic system, for instance using interval, ellipsoidal or polynomial model arithmetic, and typically perform a branch-and-bound search in the reduced space of the main decision variables. Despite much progress in recent years, their practical applicability is currently limited to problems with around 10 decision variables and small-scale dynamic systems. In contrast, direct simultaneous methods based on orthogonal collocations have been used extensively for local optimization, but seldom have they been investigated in a context of global optimization [6-7]. The simultaneous approach presents the inherent advantage that the objective and constraint functions in the discretized optimization problem are now factorable, so that in principle state-of-the-art global optimization solvers such as BARON or ANTIGONE can be applied directly. Of course, an attendant drawback is the large number of variables and constraints, which leads to prohibitive computational times but for very small problems.

The main objective of this work is to further investigate direct simultaneous methods in deterministic global optimization. Our main focus is on exploiting the underlying structure of the discretized NLP problem in order to expedite the branch-and-bound search. Specifically, the residuals from the discretization of the differential equations present a bordered-block diagonal structure, which we are exploiting to generate additional constraints between the decision variables and the state collocation variables using novel implicit equation bounders [8]. This leads to a tightening of the NLP relaxations and provides a means of being more selective with regards to the branching variable selection. We also consider various discretization schemes for the differential equations, including time-series expansion, Runge-Kutta formula, and orthogonal collocation. Comparisons are made between these improved direct simultaneous methods and existing direct sequential methods for a number of numerical case studies.

References:
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