Solving linear programs with finite precision: II. Algorithms

Abstract

We describe an algorithm that first decides whether the primal-dual pair of linear programsis feasible and in case it is, computes an optimal basis and optimal solutions. Here, A ∈ R m×n , b ∈ R m , c ∈ R n are given. Our algorithm works with finite precision arithmetic. Yet, this precision is variable and is adjusted during the algorithm. Both the finest precision required and the complexity of the algorithm depend on the dimensions n and m as well as on the condition K(A, b, c) introduced in D. Cheung and F. Cucker [Solving linear programs with finite precision: I. Condition numbers and random programs, Math. Program. 99 (2004) 175-196].

A note on level-2 condition numbers

“…When c → ∞ we have cond(1) → 0 and cond [2] (1) → 5 3 while for c = − 1 5 we have cond(1) = 5 3 and cond [2] (1) = 0.…”

“…Consider the problem of, given x ∈ R, compute f (x) = x 2 +x+c. Here, c ∈ R. Since f is continuous on R, for all x ∈ R, cond(x) = |xf (x)| |f (x)| and, assuming xf (x), f (x) > 0, cond [2] …”

“…This idea was used to define the condition number C(d) for feasibility of linear (and more general) programs [11,12], the condition number K(d) for the problem of computing an optimal basis of a linear program [1,2], or the condition number K(d) for a complementary partition problem [3]. Any problem induces another problem namely, the computation of cond .…”

“…This "condition of the condition", called level-2 condition number and denoted by cond [2] (d), was introduced by Demmel [5] where he proved, for some specific problems, that their level-2 condition numbers coincided with their original condition numbers up to a multiplicative constant. Subsequently, Higham [8] improved this result by sharpening the bounds for the problems of matrix inversion and linear systems solving.…”

A hybrid branch-and-bound approach for exact rational mixed-integer programming

“…LPs are considered ill-conditioned if small modifications in the problem data can have a large effect on the solution; in particular, if they lead to changes in the optimal objective value, changes in primal or dual feasibility, or changes in the structure of the final LP basis. Connections have been made between LP condition measures and the complexity of solving them [14,15,41,44]; ill-conditioned LPs may require higher precision arithmetic or more interior point iterations. Computational studies have also investigated these ideas [12,37].…”