(398f) Improved Relaxations for Global Optimization with ODEs and DAEs Embedded

An improved method is presented for computing convex and
concave relaxations of the parametric solutions of nonlinear ordinary
differential equations (ODEs) and nonlinear semi-explicit index-one systems of
differential-algebraic equations (DAEs).  Such relaxations,  termed state
relaxations, are fundamental to deterministic global optimization algorithms
for problems with ODEs and DAEs embedded, and their computation is the subject
of several recent articles. Largely owing to the difficulty of this computation
and the weaknesses of available methods, it remains an unfortunate fact that
state-of-the-art deterministic methods for global dynamic optimization can only
solve problems of modest size with reasonable computational effort, typically
on the order of 5 state variables and 5 decisions. On the other hand, potential
applications for such techniques are ubiquitous, including parameter estimation
problems with dynamic models, optimal control of batch processes, safety
verification problems, optimal catalyst blending , optimal drug scheduling,
etc. Moreover, representative case studies in the literature suggest that these
applications commonly lead to problems with multiple suboptimal local minima,
especially when the embedded dynamic system involves a model of chemical
reaction kinetics. Thus, the need for improved relaxation techniques is clear.

In this presentation, we study a class of state relaxation
techniques in which the desired relaxations are computed as the solutions of an
auxiliary dynamic system (either ODEs or DAEs) derived by relaxing the
governing equations of the original system in some way. Methods of this type
have the advantage that the state relaxations can be evaluated efficiently
using a state-of-that-art numerical integration code. We begin by presenting a
theoretical analysis of this class of methods that formalizes properties of the
auxiliary system that are guaranteed to lead valid state relaxations. Next, two
existing state relaxation methods for parametric ODEs are classified according
to our analysis, and it is shown that each of these methods has certain
undesirable properties that  follow directly from this classification. Based on
these observations, a new method is designed in order to avoid these deficiencies.
By numerical experiments, it is shown that the proposed method outperforms the
previous methods in terms of the tightness of the computed relaxations and the empirical
rate of convergence. Finally, the new approach is extended to the class of
semi-explicit index-one DAEs.