Minimum effort approximation of the Pareto space of convex bi-criteria problems

One of the major issues facing design practitioners using multiple-criteria optimization problems is how to decide on the trade-off between the various objectives. Since the Pareto set of those problems gives the ability to visualize the trade-off in the objective space, based on the Pareto set, a decision maker could conceiveably make trade-off decisions without repeatedly solving the problem. The Pareto set of bi-criteria problems is a curve. To generate that Pareto curve, the problem needs to be solved many times. This may not be feasible for some computationally intensive applications. One way to avoid this issue is to approximate the Pareto curve using partial information from the Pareto set. In this paper, the hyper-ellipse is used to approximate the entire Pareto set of convex bi-criteria optimization problems. The approximation is achieved by means of fitting a hyper-ellipse to a minimum number of Pareto points and the equation of the hyper-ellipse yields an explicit analytical description of the Pareto set. The method is progressively applied to unconstrained and constrained problems to understand its behavior, and illustrated on a structural example to show its efficiency. The paper identifies the limits of applicability of the approach and proposes further extensions.