Required relationship between objective function and Pareto frontier orders: Practical implications

It is well known that the weighted sum aggregate objective function fails to capture Pareto points that are located on a concave region of the Pareto frontier. Conditions are established for a general aggregate objective function to capture points on a concave Pareto frontier region. Specifically, these conditions are developed in terms of the relative orders of the Pareto frontier and aggregate objective function. These critical conditions can be used in practice to allow an aggregate objective function to capture any potentially desirable Pareto point. Conversely, failure to satisfy these critical conditions can represent a pivotal failure point of the general application of design optimization because we may fail to obtain the design we seek.