Discretize-Then-Relax Approach For State Relaxactions In Global Dynamic Optimization

This paper presents a discretize-then-relax approach to construct convex/concave bounds for the solutions of a wide class of parametric nonlinear ODEs. The procedure builds upon interval-based techniques implemented in state-of-the-art validated ODE solvers and uses McCormick's relaxation technique to propagate the convex/concave bounds. At each time step, a two-phase procedure is applied: 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. This approach is implemented in an object-oriented manner using templates and operator overloading. It is demonstrated by a case study of a Lotka-Volterra system.