Time Complexity and Zeros of the Hypervolume Indicator Gradient Field

In multi-objective optimization the hypervolume indicator is a measure for the size of the space within a reference set that is dominated by a set of μ points. It is a common performance indicator for judging the quality of Pareto front approximations. As it does not require a-priori knowledge of the Pareto front it can also be used in a straightforward manner for guiding the search for finite approximations to the Pareto front in multi-objective optimization algorithm design. In this paper we discuss properties of the gradient of the hypervolume indicator at vectors that represent approximation sets to the Pareto front. An expression for relating this gradient to the objective function values at the solutions in the approximation set and their partial derivatives is described for arbitrary dimensions m ≥ 2 as well as an algorithm to compute the gradient field efficiently based on this information. We show that in the bi-objective and tri-objective case these algorithms are asymptotically optimal with time complexity in Θ(μd+μ logμ) for d being the dimension of the search space and μ being the number of points in the approximation set. For the case of four objective functions the time complexity is shown to be in O(μd + μ). The tight computation schemes reveal fundamental structural properties of this gradient field that can be used to identify zeros of the gradient field. This paves the way for the formulation of stopping conditions and candidates for optimal approximation sets in multi-objective optimization.