We propose a methodology for optimal kinetic model reduction by simultaneous reaction and species elimination based on two integer linear programs (ILPs). The first ILP provides an optimally reduced kinetic mechanism that reproduces the predictions of a reference mechanism within prespecified tolerances for finitely many reference points in the state space. Alternatively, an integer-linear semi-infinite program is used to ensure that the production rates of the reduced mechanism are within a tolerance from the full mechanism for an entire region of the state space. Since this program is overly expensive to solve, an upper bound is obtained by overestimation of the constraints via interval extensions and the formulation of an ILP.

The need for kinetic model reduction is well established [1] and in recent years several methodologies for model reduction have been proposed, see e.g., [2, 3, 4] for reviews. Our contribution is based on the reaction elimination formulations by Green, Barton and coworkers [5, 6]. While reaction elimination has been shown to decrease computational requirements linearly with the number of reactions eliminated, species elimination has been postulated to exhibit a superlinear decrease [7]. So far either integer-nonlinear programs (INLPs) or mixed-integer linear programs (MILPs) of significantly larger size have been proposed.

The starting point for the point-constrained formulation is the ILP proposed for reaction elimination only [5] based on a kinetic mechanism which is assumed to be complete and perfect and a finite set of control points in the state space (typically obtained from a reference trajectory). The logical variables in the ILP correspond to the inclusion of a given reaction. The constraints ensure that the production rates of the reduced mechanism are within a specified tolerance from the full mechanism for all control points. Its solution is the optimally reduced kinetic mechanism. Bhattacharjee [7, p. 61-63] proposed an extension of this approach to simultaneous reaction and species elimination by introducing additional variables corresponding to the inclusion of species and additional constraints enforcing that a species is only eliminated when all reactions in which this species participates are eliminated. This new program is an INLP and as such computationally very expensive to solve. We exploit the problem structure and propose an exact reformulation as an ILP without an increase in the problem size. The size and structure of the proposed ILP are approximately the same as in the proposal of Bhattacharjee et al. [5] for reaction elimination only. Our formulation can be readily solved with standard solvers and we demonstrate its capabilities for a number of benchmark problems.

We also extend the range-constrained reaction elimination by Oluwole et al. [6] to simultaneous species and reaction elimination. In this alternative, reactions and species are eliminated while directly ensuring that the production rates are within a specified tolerance from the full mechanism for an entire region of the state space. This semi-infinite formulation is not tractable with state-of-the-art optimization algorithms and we therefore solve a rigorous conservative approximation instead. Similarly to Oluwole et al. [6], we use interval extensions obtained by Taylor models through DAEPACK [8, 9] and obtain an ILP which guarantees that the production rates are within a tolerance from the full mechanism for an entire region of the state space. The solution obtained is not guaranteed to be optimal, i.e., the number of reactions and species eliminated is potentially smaller than the maximal possible. We demonstrate the capabilities of our proposed approximation for a number of benchmark problems.

Acknowledgments

Alexander Mitsos would like to acknowledge Dr. John E. Tolsma and Dr. Glen H. Ko for fruitful discussions. Geoffrey M. Oxberry is supported by a DOE CSGF fellowship provided under grant number DE-FG02-97ER25308.

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[9] Numerica technology. www.numericatech.com.