Tight Convex and Concave Relaxations via Taylor Models for Global Dynamic Optimization

This article presents a discretize-then-relax method to construct convex/concave bounds for the solutions of parametric nonlinear ODEs. It builds upon Taylor model methods for verified ODE solution. To enable the propagation of convex/concave state bounds, a new type of Taylor model is introduced, whereby the remainder term consists of convex/concave bounds in lieu of the usual interval bounds. At each time step, a two-phase procedure is applied for the verified integration. A priori convex/concave bounds that are valid over the entire time step are calculated in the first phase, then pointwise-in-time convex/concave bounds at the end of the time step are obtained in the second phase. The algorithm is demonstrated by the case study of a Lotka-Volterra system.