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

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.