Optimal bounding of curves by continuous piecewise linear functions

This paper presents a numerical method to solve a class of optimization problems which is defined as follows. Bound a design curve (which may be dependent on some system parameters) from above by a continuous piecewise linear function of a given number of segments such that the area under the function has a minimum value. The decision variables include the system parameters and the parameters that describe the piecewise linear function. For illustrative purposes, the numerical method is applied to two optimal design problems of linear segmented beams. In the case of a convex design curve, a simpler geometry-based algorithm is presented for solution.