Monotonicity versus performance in co-optimization

• Authors: Elena Popovici;Kenneth De Jong
• Venue: Proceedings of the tenth ACM SIGEVO workshop on Foundations of genetic algorithms
• Year: 2009

Applying evolutionary algorithms to traditional single-objective optimization problems is now a well-understood and successful process. Less well understood but developing nicely is the use of evolutionary algorithms for multi-objective optimization problems. Least well understood is the use of coevolutionary algorithms to solve co-optimization problems. The dynamics of these coevolutionary processes lead to pathological situations not seen (or easily avoided) in standard optimization algorithms. Such pathologies include endless cycling, loss of fitness gradient, and so on. These problems are due in part to the fact that in many coevolutionary settings even basic notions such as what the goal is or what it means to make progress toward the goal are often vague and ill-defined. Thorough characterizations of possible goals for co-optimization have been introduced and we now have a fairly broad range of so-called ``solution concepts''. Once the idea of a goal was ironed out, the bulk of co-optimization research to date has focused on defining and achieving monotonic progress toward the chosen goal. Via careful analysis and generalizations, this paper reconciles different and apparently conflicting notions of monotonicity existent in the field. However, this analysis also points out that even the weakest monotonicity may simply not be within reach for some solution concepts that are nonetheless of practical interest. Moreover, the monotonicity results tell us nothing about the relative performance of different algorithms. The hypothesis put forward, in light of recent ``free lunch'' results, is that we may be faced with a difficult choice between monotonicity and performance. These ideas are formally developed and analyzed in the context of standard solution concepts such as maximum cumulative utility and best worst case.