Inner Approximations of Consistent Parameter Sets by Constraint Inversion and Mixed-Integer Programming

Mathematical modeling has become an indispensable tool in the analysis, prediction and control of chemical and biological systems. However, the estimation of consistent model parametrizations and model invalidation are challenging tasks, but crucial for reliable model-based analysis and prediction. Set-based estimation methods are useful to determine guaranteed outer approximations of consistent parameter sets, i. e. consistent parametrizations are never excluded. However, these conservative outer approximating sets often include inconsistent parametrizations which lead to inconsistent models and hence possibly wrong model-based predictions. This paper proposes a set-based framework to determine inner approximations, i.e. the model is guaranteed consistent with measurement data for all parametrizations from this set. Our approach is based on the reformulation and inversion of measurement data constraints and by imposing nonlinear constraints on binary variables. The relaxation of the mixed-integer nonlinear feasibility problem into a mixed-integer linear feasibility problem allows the inner approximations to be determined efficiently. The applicability of this approach is demonstrated considering a nonlinear biochemical reaction network.