The Discrete Moment Method for the numerical integration of piecewise higher order convex functions

A new numerical integration method, termed Discrete Moment Method (DMM), is proposed for univariate functions that are piecewise higher order convex. This means that the interval where the function is defined can be subdivided into non-overlapping subintervals such that in each interval all divided differences of given orders, do not change the sign. The new method uses piecewise polynomial lower and upper bounds on the function, created in connection with suitable dual feasible bases in the univariate discrete moment problem and the integral of the function is approximated by tight lower and upper bounds on them. Numerical illustrations are presented for the cases of the normal, exponential, gamma and Weibull probability density functions.