Which Point From an Interval Should We Choose?

In many practical situations, we know the exact form of the objective function, and we know the optimal decision corresponding to each value of the corresponding parameter x. What should we do if we do not know the exact value of x, and instead, we only know x with uncertainty - e.g., with interval uncertainty? In this case, a reasonable idea is to select one value from the given interval, and to use the optimal decision corresponding to the selected value. But which value should we choose? In this paper, we provide a solution to this problem for the situation in the simplest 1-D case. Somewhat surprisingly, it turns out the usual practice of selecting the midpoint is rarely optimal, a better selection is possible.