A geometric view of parametric linear programming

Abstract

We present a new definition of optimality intervals for the parametric right-hand side linear programming (parametric RHS LP) Problem ϑ(λ) = min{c ^t x¦Ax =b + λ¯b,x ≥ 0}. We then show that an optimality interval consists either of a breakpoint or the open interval between two consecutive breakpoints of the continuous piecewise linear convex function ϑ(λ). As a consequence, the optimality intervals form a partition of the closed interval {λ; ¦ϑ(λ)¦ < ∞}. Based on these optimality intervals, we also introduce an algorithm for solving the parametric RHS LP problem which requires an LP solver as a subroutine. If a polynomial-time LP solver is used to implement this subroutine, we obtain a substantial improvement on the complexity of those parametric RHS LP instances which exhibit degeneracy. When the number of breakpoints of ϑ(λ) is polynomial in terms of the size of the parametric problem, we show that the latter can be solved in polynomial time.