Understanding Knee Points in Bicriteria Problems and Their Implications as Preferred Solution Principles

A knee point is almost always a preferred trade-off solution, if it exists in a bicriteria optimization problem. In this paper, we make an attempt to better understand knee points and investigate the properties of bicriteria problems that may exhibit a knee on its Pareto-optimal front. Specifically, we review past studies and suggest a couple of new definitions of a knee point. One of the definitions allows us to define a knee region for problems in which, instead of one, a set of knee-like solutions exist. We also introduce edge-knee solutions which behave like a knee solution but lies near one of the extreme solutions. We argue here that knee solutions have far-reaching implications than they are currently known for. It is an interesting fact that in many problem-solving tasks, despite the existence of a number of solution methodologies, one or a few methodologies are commonly used. Here, we argue that often such common solution principles are knee solutions to a bicriteria problem formed from two conflicting goals of the underlying problem-solving task. We illustrate our argument by show-casing a number of popularly used problem-solving tasks, such as regression, sorting, clustering and a number of engineering design tasks. Each task, when viewed as a bicriteria problem, seems to exhibit a knee or a knee-region and the commonly used methodology seems to lie within the knee-region. Linking preferred solution methodologies as knee solutions to a related bicriteria problem is certainly an interesting idea and may have a long-term implication in the development of efficient solution methodologies for different scientific and other problem-solving tasks, such as game-playing and strategic decision-making activities.