I-SMOOTH: Iteratively Smoothing Mean-Constrained and Nonnegative Piecewise-Constant Functions

  • Authors:

  • Huifen Chen;Bruce Schmeiser

  • Affiliations:

  • Department of Industrial and Systems Engineering, Chung-Yuan University, Chung-Li, Taoyuan, 320, Taiwan;School of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907/ and Department of Industrial and Systems Engineering, Chung-Yuan University, Chung-Li, Taoyuan, 320, Taiwan

  • Venue:
  • INFORMS Journal on Computing
  • Year:
  • 2013

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Abstract

Continuous nonnegative functions, such as Poisson rate functions, are sometimes approximated as piecewise-constant functions. We consider the problem of automatically smoothing such functions while maintaining the integral of each piece and maintaining nonnegativity everywhere, without specifying a parametric function. We develop logic for SMOOTH Smoothing via Mean-constrained Optimized-Objective Time Halving, a quadratic-optimization algorithm that yields a smoother nonnegative piecewise-constant rate function having twice as many time intervals, each of half the length. I-SMOOTH Iterated SMOOTH iterates the SMOOTH formulation to create a sequence of piecewise-constant rate functions that, in the limit, yields a nonparametric continuous function. We consider two contexts: finite-horizon and cyclic. We develop a sequence of computational simplifications for SMOOTH, moving from numerically minimizing the quadratic objective function, to numerically computing a matrix inverse, to a closed-form matrix inverse obtained as finite sums, to optimal decision-variable values that are linear combinations of the given rates, and to simple approximations.