A canonical form for pencils of matrices with applications to asymptotic linear programs Original Research Article

This paper studies asymptotic pencils of linear programs in which the constraint matrix, right-hand sides and objective function coefficients depend linearly on a parameter, and we seek a basis that is optimal for all large (or small) enough values of the parameter. The asymptotic linear programming formulation of discounted Markov decision chains has this form, as do many other problems. For this problem, we introduce a new canonical form for bases that facilitates computation of Laurent expansions of the solutions of the resulting asymptotic linear programs and show how to implement the simplex method efficiently in this setting. We show that the computational work to solve the asymptotic linear program is the product of the number of rows of the problem and the work required to solve an ordinary linear program of the same size. This improves computational efficiency of the asymptotic simplex method of Jeroslow (1973) and a related algorithm considered by Lamond (1989). In essence this work generalizes that of Miller and Veinott (1969) and Rothblum (1974, 1975) for Markov ordinary and Markov branching decision chains to arbitrary linear programs.