Solving multiscale linear programs using the simplex method in quadruple precision

Abstract

Systems biologists are developing increasingly large models of metabolism and integrated models of metabolism and macromolecular expression. These Metabolic Expression (ME) models lead to sequences of multiscale linear programs for which small solution values of order 10−6 to 10−10 are meaningful. Standard LP solvers do not give sufficiently accurate solutions, and exact simplex solvers are extremely slow.We investigate whether double-precision and quadrupleprecision simplex solvers can together achieve reliability at acceptable cost. A double-precision LP solver often provides a reasonably good starting point for a Quad simplex solver. On a range of multiscale examples we find that 34-digit Quad floating-point achieves exceptionally small primal and dual infeasibilities (of order 10−30) when no more than 10−15 is requested. On a significant ME model we also observe robustness in almost all (even small) solution values following relative perturbations of order 10−6 to non-integer data values. Double and Quad Fortran 77 implementations of the linear and nonlinear optimization solver MINOS are available upon request.