Abstract-The goal of multiobjective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most real-life problems, only an approximation to such an optimal set can be obtained within reasonable (computing) time. To compare such approximations, and thereby the performance of multiobjective optimizers providing them, unary quality measures are usually applied. Among these, the hypervolume indicator (or S-metric) is of particular relevance due to its favorable properties. Moreover, this indicator has been successfully integrated into stochastic optimizers, such as evolutionary algorithms, where it serves as a guidance criterion for finding good approximations to the Pareto front. Recent results show that computing the hypervolume indicator can be seen as solving a specialized version of Klee's Measure Problem. In general, Klee's Measure Problem can be solved with O(n log n + n d/2 log n) comparisons for an input instance of size n in d dimensions; as of this writing, it is unknown whether a lower bound higher than (n log n) can be proven. In this paper, we derive a lower bound of (n log n) for the complexity of computing the hypervolume indicator in any number of dimensions d > 1 by reducing the so-called UNIFORMGAP problem to it. For the 3-D case, we also present a matching upper bound of O(n log n) comparisons that is obtained by extending an algorithm for finding the maxima of a point set. Index Terms-Complexity analysis, computational geometry, multiobjective optimization, performance assessment. I. MOTIVATION AND INTRODUCTION TO MULTIOBJECTIVE OPTIMIZATION N MULTIOBJECTIVE optimization, the problem is to find best possible compromise solutions which cannot be improved according to one objective without deteriorating the others. This type of problem arises in all kind of industrial application areas ranging from production to service industries, entertainment, and many others. However, since many realworld problems cannot be expected to be solved to optimality (whether at all or within a reasonable amount of computing time), the goal is usually to obtain a good approximation to the optimal set of solutions within a reasonable amount of time.  e., q p) and q = p. Two distinct points are incomparable iff neither point dominates the other. Points that are not dominated within a set are the best ones, and are referred to as non-dominated or maximal. The elements of the search space that generate the non-dominated elements of the objective space form the Pareto set of the problem, and the set of the corresponding images in the objective space is called Pareto front. Multiobjective optimizers generate approximations of the Pareto front. To assess the performance of different optimizers, their resulting approximations have to be compared. This may be performed by extending the Pareto dominance relation to sets of points (see e.g., Zitzler et al. [3]), but, in this case, good Pareto front approximations are often incomparable to one another. Therefore, many researchers have proposed quality indicators for the sets of compromise solutions generated by multiobjective optimizers that-according to several quality criteria-map such sets onto scalar values and thus allow for an easy comparison. There is a general consensus about two (informal) criteria of quality: An approximation of the optimal set is good if 1) its points are "close" to the Pareto front; 2) the points are "welldistributed" along the whole Pareto front. Additionally, we find it worth mentioning that a good approximation shall contain "many" non-dominated points. An in-depth overview of quality measures and their properties is given by Zitzler et al. [3].
For example, we showed that QHV obtains O(n log n) time, in the best case, which is the lower bound REF for hypervolume but this performance is virtually impossible.
On the Complexity of Computing the Hypervolume Indicator
IEEE Transactions on Evolutionary Computation
IEEE Transactions on Evolutionary Computation
Mathematics Computer Science