An O(n) algorithm for discrete n-point convex approximation with applications to continuous case

Given a bounded real function ? defined on a closed bounded real interval I, the problem is to find a convex function g so as to minimize the supremum of $\text{¦f(t) ? g(t)¦}$ for all t in I, over the class of all convex functions on I. The usual approach is to consider a discrete version of the problem on a grid of (n + 1) points in I, apply a conventional linear program to obtain an optimal solution, and let the grid size go to zero. This paper presents an alternative algorithm of complexity O(n), which is based on the concept of the greatest convex minorant of a function, for computation of a special “maximal” optimal solution to the discrete problem. It establishes the rate of convergence of this optimal solution to a solution of the original problem as the grid size goes to zero. It presents an alternative efficient linear program that generates the maximal optimal solution to the discrete problem. It also gives an O(n) algorithm for the discrete n-point monotone approximation problem.