Equivalence of linear deviation about the mean and mean absolute deviation about the mean objective functions

Konno, introduced a piecewise linear objective function for portfolio optimization to measure the deviation from a mean return. An apparently asymmetric objective function can be obtained by changing the gradients either side of the mean. However, we show that when the linear deviations are taken relative to the mean, any two piece linear objective function is equivalent to the mean absolute deviation, which is symmetric. Equivalent is used here to mean that one function is proportional to the other. Also we show that emphasizing upside risk is exactly equal to emphasizing downside risk when these are taken relative to the mean. No distributional assumptions are required beyond the existence of the first moment. In this case an investor changing from upside to downside risk would not change his solution at all, despite what the investor intended to achieve.