Certified Scalable Enclosures for Uncertain Underdetermined Systems

The central challenge in underdetermined models with bounded uncertainty, such as in nonlinear design and estimation, is certifying the solution sets. Conventional solution methodologies, such as those based on Newton's method or on sampling-based uncertainty quantification, are either not applicable due to the underdetermined nature or do not give guarantees that all solutions have been found, assuming they converge. We addressed this issue for a problem that arises in nonlinear design, where a target steady-state box is prescribed and the parameters have to be found. We developed a trapezoidal linearisation method that rigorously encloses all solutions and combined it with a tractable linear programming method to compute the component-wise bounds. We showed that the resulting families of linear programs can iteratively contract an initial parameter region. A subdivision of the target state and the initial parameter region can improve the contraction and provide tighter enclosures. The trapezoidal relaxation gives a family of linear programs that can also be used for finding solutions for interval linear systems via the Oettli-Prager characterisation. We demonstrate the generality of the method through different applications, including nonlinear design in biomolecular circuits, a sensitivity analysis method, and in a compressed sensing context, and discuss the trade-off between enclosure tightness and computational cost.