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**11 - 14**of**14**### A Review of Multiobjective Test Problems and a Scalable Test Problem Toolkit

"... Abstract—When attempting to better understand the strengths and weaknesses of an algorithm, it is important to have a strong understanding of the problem at hand. This is true for the field of multiobjective evolutionary algorithms (EAs) as it is for any other field. Many of the multiobjective test ..."

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Abstract—When attempting to better understand the strengths and weaknesses of an algorithm, it is important to have a strong understanding of the problem at hand. This is true for the field of multiobjective evolutionary algorithms (EAs) as it is for any other field. Many of the multiobjective test problems employed in the EA literature have not been rigorously analyzed, which makes it difficult to draw accurate conclusions about the strengths and weaknesses of the algorithms tested on them. In this paper, we systematically review and analyze many problems from the EA literature, each belonging to the important class of real-valued, unconstrained, multiobjective test problems. To support this, we first introduce a set of test problem criteria, which are in turn supported by a set of definitions. Our analysis of test problems highlights a number of areas requiring attention. Not only are many test problems poorly constructed but also the important class of nonseparable problems, particularly nonseparable multimodal problems, is poorly represented. Motivated by these findings, we present a flexible toolkit for constructing well-designed test problems. We also present empirical results demonstrating how the toolkit can be used to test an optimizer in ways that existing test suites do not. Index Terms—Evolutionary algorithms (EAs), multiobjective evolutionary algorithms, multiobjective optimization, multiobjective test problems. I.

### Computing Klee’s Measure of Grounded Boxes

- ALGORITHMICA
, 2012

"... A well-known problem in computational geometry is Klee’s measure problem, which asks for the volume of a union of axis-aligned boxes in d-space. In this paper, we consider Klee’s measure problem for the special case where a 2-dimensional orthogonal projection of all the boxes has a common corner. ..."

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A well-known problem in computational geometry is Klee’s measure problem, which asks for the volume of a union of axis-aligned boxes in d-space. In this paper, we consider Klee’s measure problem for the special case where a 2-dimensional orthogonal projection of all the boxes has a common corner. We call such a set of boxes 2-grounded and, more generally, a set of boxes is k-grounded if in a k-dimensional orthogonal projection they share a common corner. Our main result is anO(n(d−1)/2 log2 n) time algorithm for computing Klee’s mea-sure for a set of n 2-grounded boxes. This is an improvement of roughly O( n) compared to the fastest solution of the general problem. The algorithm works for k-grounded boxes, for any k ≥ 2, and in the special case of k = d, also called the hypervolume indicator problem, the time bound can be improved further by a logn factor. The key idea of our technique is to reduce the d-dimensional problem to a semi-dynamic weighted volume problem in dimension d − 2. The weighted volume problem requires solving a combinatorial problem of maintaining the sum of ordered products, which may be of independent interest.