Linear time algorithms for convex and monotone approximation

Abstract

This article presents algorithms of linear time complexity ( O( n)) for computation of optimal solutions to the two problems of convex and monotone approximation where data points are approximated, respectively, by convex and monotone (nondecreasing) functions on a grid of ( n + 1) points. For the convex approximation case, the algorithms are based on a linear programming approach which exploits the structure of matrices involved and uses a special pivoting procedure to obtain a “maximal” optimal solution. Analogously, in the monotone approximation case, the algorithms compute “maximal” and “minimal” optimal solutions which also “enclose” any other optimal solution between them. Computational results are presented.