On characterizing the "knee" of the Pareto curve based on Normal-Boundary Intersection

This paper deals with the issue of generating one Pareto optimal point that is guaranteed to be in a "desirable" part of the Pareto set in a given mul ticriteria optimization problem. A parameterization of the Pareto set based on the recently developed normal-boundary intersection technique is used t o formulate a subproblem, the solution of which yields the point of "maximu m bulge", often referred to as the "knee of the Pareto curve". This enables the identification of the "good region" of the Pareto set by solving one n onlinear programming problem, thereby bypassing the need to generate many P areto points. Further, this representation extends the concept of the "knee " for problems with more than two objectives. Tt is further proved that thi s knee is invariant with respect to the scales of the multiple objective fu nctions. The generation of this knee however requires the value of each objective fu nction at the minimizer of every objective function (the pay-off matrix). T he paper characterizes situations when approximations to the function value s comprising the pay-off matrix would suffice in generating a good approxim ation to the knee. Numerical results are provided to illustrate this point. Further, a weighted sum minimization problem is developed based on the inf ormation in the pay-off matrix, by solving which the knee can be obtained.