Hypervolume indicator is a commonly accepted quality measure for comparing Pareto approximation set generated by multi-objective optimizers. The best known algorithm to calculate it for $n$ points in $d$-dimensional space has a run time of $O(n^{d/2})$ with special data structures. This paper presents a recursive, vertex-splitting algorithm for calculating the hypervolume indicator of a set of $n$ non-comparable points in $d>2$ dimensions. It splits out multiple child hyper-cuboids which can not be dominated by a splitting reference point. In special, the splitting reference point is carefully chosen to minimize the number of points in the child hyper-cuboids. The complexity analysis shows that the proposed algorithm achieves $O((\frac{d}{2})^n)$ time and $O(dn^2)$ space complexity in the worst case.
Optimization for multiple conflicting objectives results in more than one optimal solutions (known as Pareto-optimal solutions). Although one of these solutions is to be chosen at the end, the recent trend in evolutionary and classical multiobjective optimization studies have focused on approximating the set of Paretooptimal solutions. However, to assess the quality of Pareto approximation set, special measures are needed [1].
Hypervolume indicator is a commonly accepted quality measure for comparing approximation set generated by multi-objective optimizers. The indicator measures the hypervolume of the dominated portion of the objective space by Pareto approximation set and has received more and more attention in recent years [2,3,1,4].
There have been some studies that discuss the issue of fast hypervolume calculation [5][6][7][8]. These algorithms partition the covered space into many cuboidshaped regions, within which the approach considering the dominated hypervolume as a special case of Klee’s measure problem is regarded as the current best one. This approach [8] adopts orthogonal partition tree which requires O(n d/2 ) storage and streaming variant [9]. Conceptual simplification of the implementation are concerned and thus the algorithm achieves an upper bound of O(n log n + n d/2 ) for the hypervolume calculation. Ignoring the running time of sorting the points according to the d-th dimension, O(n log n), the running time of this approach is exponential of the dimension of space d.
This paper develops novel heuristics for the calculation of hypervolume indicator. Special technologies are applied and the novel approach yields upper bound of O(( d 2 ) n ) runtime and consumes O(dn 2 ) storage. The paper is organized as follows. In the next section, the hypervolume indicator is defined, and some background on its calculation is provided. Then, an algorithm is proposed which uses the so-called vertex-splitting technology to reduce the hypervolume. The complexities of the proposed algorithm are analyzed in Section 4. The last section concludes this paper with an open problem.
Without loss of generality, for multi-objective optimization problems, if the d objective functions f = (f 1 , . . . , f d ) are considered with f i to be minimized, not one optimal solution but a set of good compromise solutions are obtained since that the objectives are commonly conflicting. The compromise solutions are commonly called Pareto approximation solutions and the set of them is called the Pareto approximation set. For a Pareto approximation set M = {y 1 , y 2 , . . . , y n } produced in a run of a multi-objective optimizer, where y i = (y i1 , . . . , y id ) ∈ M ⊂ R d , all the solutions are non-comparable following the well-known concept of Pareto dominance. Specially, we say that y i dominates y k at the j-th dimension if y ij < y kj .
The unary hypervolume indicator of a set M consists of the measure of the region which is simultaneously dominated by M and bounded above by a reference point r = (r 1 , . . . , r d ) ∈ R d such that r j ≥ max i=1,…,n {y ij }. In the context of hypervolume indicator, we call the solutions in M as the dominative points. As illustrated in Fig. 1(a), the shading region consists of an orthogonal polytope, and may be seen as the union of three axis-aligned hyperrectangles with one common vertex, i.e., the reference point r. Another example in three dimensional space is shown in Fig. 1(b), where five dominative points, y 1 = (1, 2, 3), y 2 = (4, 3, 2), y 3 = (5, 1, 4), y 4 = (3, 5, 1), y 5 = (2, 2, 2.5), and the reference point r = (6, 6, 6) are considered. The volume is the union of the volumes of all the cuboids each of which is bounded by a vertex, where the common regions are counted only once. If a point y k is dominated by another point y i , the cuboid bounded by y k is completely covered by the cuboid bounded by y i . And thus only the non-dominated points contribute to the hypervolume.
In other works, e.g. the work of Beume and Rudolph [8], the hyper-cuboid in d-dimensional space are partitioned into child hyper-cuboids along the d-th dimension and then all these child hypervolumes are gathered together by the inclusion-exclusion principle [10]. In this paper, we step in another way. The hyper-cuboid is partitioned into child hyper-cuboids at some splitting reference points and then all the child hypervolumes are gathered directly. More detailed, given a point y i ∈ M , each of other points in M must dominated y i at some dimensions for the non-comparable relation. If the parts over y i are handled, the problem of calculating the hypervolume bounded by M and the reference point is figured out. The additional part partitioned out at the j-th dimension is also a d-dimensional hyper-cuboid whose vertices are ones beyond y i at such dimension. Their projections on the hyperplane orthogonal to dimension j are all dominated by y i , and thus are free from consideration. It should b
This content is AI-processed based on open access ArXiv data.