The Isotonic Regression Problem and its Dual

The isotonic regression problem is to minimize Σ t i = 1 [g i − x i ] 2 wi subject to x i ≤ x j when where w i >0 and g i (i= 1, 2, ..., k) are given and is a specified partial ordering on {1, 2, ..., k}. The solution is called the isotonic regression on g. We formulate a generalization of this problem and calculate its Fenchel dual. A function of the isotonic regression also solves these problems. Problems in inventory theory and statistics are identified as dual isotonic regression problems.