Discretize-then-relax approach for convex/concave relaxations of the solutions of parametric ODEs

This paper presents a discretize-then-relax methodology to compute convex/concave bounds for the solutions of a wide class of parametric nonlinear ODEs. The procedure builds upon interval methods for ODEs and uses the McCormick relaxation technique to propagate convex/concave bounds. At each integration step, a two-phase procedure is applied: a priori convex/concave bounds that are valid over the entire step are calculated in the first phase; then, pointwise-in-time convex/concave bounds at the end of the step are obtained in the second phase. An approach that refines the interval state bounds by considering subgradients and affine relaxations at a number of reference parameter values is also presented. The discretize-then-relax method is implemented in an object-oriented manner and is demonstrated using several numerical examples.