Minimizing a Convex Cost Closure Set

Many applications in the area of production and statistical estimation are problems of convex optimization subject to ranking constraints that represent a given partial order. This problem – which we call the convex cost closure problem, or (CCC) – is a generalization of the known maximum (or minimum) closure problem and the isotonic regression problem. For a (CCC) problem on n variables and m constraints we describe an algorithm that has the complexity of the minimum cut problem plus the complexity of finding the minima of up to n convex functions. Since the convex cost closure problem is a generalization of both minimum cut and of minimization of n convex functions this complexity is fastest possible. For the quadratic problem the complexity of our algorithm is strongly polynomial, O(mn log n 2 m ). For the isotonic regression problem the complexity is O(n log U), for U the largest range for a variable value.