PaCcET: An Objective Space Transformation to Iteratively Convexify the Pareto Front

In multi-objective problems, it is desirable to use a fast algorithm that gains coverage over large parts of the Pareto front. The simplest multi-objective method is a linear combination of objectives given to a single-objective optimizer. However, it is proven that this method cannot support solutions on the concave areas of the Pareto front: one of the points on the convex parts of the Pareto front or an extreme solution is always more desirable to an optimizer. This is a significant drawback of the linear combination. In this work we provide the Pareto Concavity Elimination Transformation (PaCcET), a novel, iterative objective space transformation that allows a linear combination (in this transformed objective space) to find solutions on concave areas of the Pareto front (in the original objective space). The transformation ensures that an optimizer will always value a non-dominated solution over any dominated solution, and can be used by any single-objective optimizer. We demonstrate the efficacy of this method in two multi-objective benchmark problems with known concave Pareto fronts. Instead of the poor coverage created by a simple linear sum, PaCcET produces a superior spread across the Pareto front, including concave areas, similar to those discovered by more computationally-expensive multiobjective algorithms like SPEA2 and NSGA-II.