Error Analysis for Convex Separable Programs: The Piecewise Linear Approximation and The Bounds on The Optimal Objective Value

The bounds have been established on the possible deviation of the optimal objective value of a separable, convex program from the optimal objective value of a program which is its piecewise linear approximation based on a given subdivision interval. By a separable, convex program is meant a separable program with proper convexity-concavity properties which imply that any local optimum is also a global optimum. It is further shown that these bounds are actually attainable, and therefore, cannot be improved in general. Some examples are provided.Naturally, the inquiry requires some study of the piecewise linear approximation itself. The bounds on the function error are determined based on the assumptions—the boundedness of the first, or the second, or both the derivatives—about the original function. Some results are derived for a given piecewise linear function, to determine the nature of the original function with minimum error and satisfying certain conditions; these results would be applicable in those ...