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The isotonic median regression problem arises in statistics. It is known that the isotonic median regression problem, with respect to a complete order, may be solved by a “Pool Adjacent Violators” algorithm. In this paper we show that this algorithm is a dual method for solving a linear programming formulation of the problem. The linear programming approach provides additional insight into the algorithm as well as a simple proof of its validity. We also analyze the computational complexity of the algorithm and discuss its significance from the standpoint of linear programming theory.

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We present algorithms to calculate the stability radius of optimal or approximate solutions of binary programming problems with a min sum or min max objective function. Our algorithms run in polynomial time if the optimization problem itself is polynomially solvable. We also extend our results to the tolerance approach to sensitivity analysis.