Computational complexity of parametric linear programming

where A is a matrix of order m × p and rank m, and A is a real valued parameter. It may be required to solve this problem for all real values of A, or for all values of A in some specified interval of the real line. f(A) is a piecewise linear convex function in A. The simplex algorithm for this parametric linear program partitions the real line into intervals, each interval being the optimality interval of a feasible basis for (1), and the algorithm moves f rom one interval to the next by making a single dual simplex pivot step. In each interval the slope o f / ( A ) is constant. In consecut ive intervals obtained during the algorithm, the slope o f / ( A ) remains the same if the algorithm moves from one of these intervals to the next without making any nondegenerate dual simplex pivot steps; otherwise the slopes o f / ( A ) in these intervals are different. See Chapters 11, 12 in [1], Chapter 8 in [2] and Chapter 7 in [3]. Let 4,(A, b, b*, c) denote the total number of intervals on the real line, each of positive length, such that the slope o f / ( A ) is a constant in each interval, and the slopes of f(A) in any two intervals are different. Since each of these intervals have to be separately obtained (as the slopes of f(A) in them are different) when (1) is solved, ~b(A, b, b*, c) provides a lower bound on the computational effort required for solving (1) for all real values of A by any algorithm. Clearly, the