Computational modeling of metabolism and macromolecular synthesis requires numerical simulation of reaction rates that span many orders of magnitude. For instance, Flux Balance Analysis of Metabolic Expression models leads to multiscale linear programming (LP) problems in which both data values and solution values have greatly varying magnitudes. Some data values are integers of order \(10^{6}\), and meaningful solution values can be as large as \(10^{8}\) or as small as \(10^{-10}\). Standard LP solvers may not give sufficiently accurate solutions, and may have difficulty determining whether a feasible solution exists. Systems biologists have applied exact simplex solvers, but they are extremely slow on large models, and model sizes will continually increase. We investigate whether double-precision and quadruple-precision simplex solvers can together achieve reliability with acceptable efficiency.

We developed Double and Quad versions of our linear and nonlinear optimization solver MINOS and employed them in tandem on a range of challenging LP models. The simplex method in Double MINOS usually gives a good starting point for the same simplex method in Quad MINOS. Hence, much of the work can be performed efficiently by Double MINOS with conventional 16-digit floating-point hardware to obtain near-optimal solutions. For Quad MINOS, 34-digit floating-point operations are implemented in the compiler’s Quad math library via software (on today’s machines). Each simplex iteration is therefore considerably slower, but the reward is extremely high accuracy. On a range of multiscale problems we achieve primal and dual infeasibilities as small as \(10^{-30}\) when just \(10^{-15}\) is requested. On a significant Metabolic Expression model we also observe robustness in almost all (even small) solution values following relative perturbations of order \(10^{-6}\) to non-integer data values.