A Geometric View of Parametric Linear Programming 1

We present a new definition of optimality intervals for the parametric right-hand side linear programming (parametric RHS LP) Problem r = m i n { c r x l A x = b + 2b, x > 0}. We then show that an optimality interval consists eifher of a breakpoint or the open interval between two consecutive breakpoints of the continuous piecewise linear convex function ~o(2). As a consequence, the optimality intervals form a partition of the closed interval {2; 1~0(2)1 < oo}. 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 q~(2) is polynomial in terms of the size of the parametric problem, we show that the latter can be solved in polynomial time.