An efficient basis update for asymptotic linear programming

Abstract

For a linear program in which the constraint coefficients vary linearly with the time parameter, we showed in a previous paper that a basic feasible solution can be evaluated using O(( k + 1) m 3) arithmetic operations, where m is the number of constraints and k is the index of the basis matrix pair. Here we show, in the special case when k = 1 for all basis matrix pairs, and when one of the matrices in each pair has nearly full rank, how the (possibly singular) matrix factorization can be updated with only O( m 2) operations, using rank-one update techniques. This makes the arithmetic complexity of updating the basis in asymptotic linear programming comparable to that of updating the inverse in ordinary linear programming, in this case. Moreover, we show that the result holds, in particular, when computing a Blackwell optimal policy for Markov decision chains in the unichain case or when all policies have only a small number of recurrent subchains.