An Inequality Sorting Algorithm for a Class of Linear Programming Problems

The linear programming problem (lpp) considered here is Maximize btx subject to the inequalities c ≤ Ax ≤ d, where A is an m × n real matrix of rank r ≤ m, the vectors c and d are real, and d ≤ 0, and the lpp is feasible with bounded optimal solutions. An algorithm that normalizes the inequalities, arranges them in an ascending order on the right-most (rm) quantities, and then chooses r linearly independent inequalities is described. It computes a solution of the lpp with these r inequalities that (the solution) determines which inequalities are not satisfied on the left-most (lm) inequalities. The unsatisfied inequalities are then negated and the algorithm is executed for the second time, This process is continued until all the inequalities are satisfied. The algorithm is, in practice, amazingly fast, often direct with computational complexity O(mn2). An application of this algorithm to a general 1pp is discussed,