269119 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.