The Isotonic Regression Problem and Its Dual

The isotonic regression problem is to minimize Σt i = 1 [gi − xi]2wi subject to xi ≤ xj when where wi>0 and gi (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.