Selection Rules

In the coverage-based Box-Algorithm, the next box for investigation is selected due to some selection rule (see line 5 in Algorithm 5.1). As mentioned at the beginning of Section 5.3, for instance, the box with the largest corner point distance can be selected, i.e., in a practical implementation, the set S can be implemented as a list for which the elements are ordered according to their corner point distances. We call this selection rule the max-dist selection rule. The max-dist selection rule is helpful if the algorithm should be aborted prematurely (e.g., by aiming at the cardinality of the representative system) since, in this case, a guaranteed bound for the obtained coverage error can be stated.

Corollary 5.38: For Algorithm 5.1, let the subroutine SelectBox() always select

the box with largest corner point distance (max-dist selection rule). Suppose the algorithm is aborted prematurely after Γ ≥ 1 iterations and for all remaining boxes B ∈ S, we additionally invoke the completion step, i.e., line 8 to 11 from Algorithm 5.1. Then, the algorithm outputs a collection of boxes B containing all nondominated points. At termination, the representative system Rep has a coverage error of at most L · 2−b(log7(6Γ+1))/2c _{(w.r.t. k · k}

∞), where L equals the corner point distance k`0− u0k∞

of the initial box B(`0, u0).

Proof:

Due to the construction of the boxes, a subdivision is obtained which contains Y_N at each state of the algorithm (see Lemma 5.14). Analogous to the proof of Theorem 5.15, we consider the tree for which each node corresponds to a box in some iteration. Due to the utilized selection rule and the proof of Theorem 5.15, we have: If the Box-Algorithm selects a box from level 2k + 1 or higher, k ∈ N0, then all remaining

boxes (including the selected box) have a corner point distance of at most L_/2k_{. From}

level 1 to level 2k, the number of nodes in the tree is bounded by 2k−1 X i=0 7i= 7 2k_{− 1} 6 .

Let k∗_{∈ N}0 be the largest number fulfilling 72k∗− 1

6 ≤ Γ .

This means, in some iteration less than or equal to Γ, the algorithm has already selected a box corresponding to a node at a level 2k∗+ 1 or higher, or, without the premature abortion, in iteration Γ + 1, the algorithm would select a box corresponding to a node at a level 2k∗+ 1 or higher. Thus, at termination, all boxes have a corner point distance of at most

L

2k∗ = L · 2

−b(log7(6Γ+1))/2c_.

 However, selecting the box with maximal corner point distance can lead to the generation of a lot of dominated solutions since a found solution can be dominated by a new solution which is found in a later iteration. In the following, we outline the idea of a new selection rule, called nondominated selection rule, for which a clever sorting of the list S of unexplored boxes is chosen in such a way that the output Rep contains only few or even no points which are dominated by other points in Rep. We assume here the incorporation of the deletion of dominated as well as partially dominated boxes as described in Section 5.8.1. Furthermore, we assume that SelectBox(S) is implemented in such a way that it always selects the first element of the given list S. Consider an arbitrary iteration of Algorithm 5.1, w.l.o.g. after an outcome z∗ was found by solving problem (P_ε1

1,ε2) (see line 19). Let B

1,1 _{to B}1,7 _{denote the new boxes.} In the modified algorithm, a new box is not added to the list S by looking at its corner point distance but with respect to some dominance relation. No point in the interior of the boxes B1,2, B1,3 and B1,6 can be dominated by a point in any other of the remaining six boxes. Hence, these three boxes (w.l.o.g. in the given ordering) are added at the beginning of the list S, which defines the ordering of the next selected boxes. Thereafter, box B1,1 is added since all points in its interior can only be dominated by points from B1,2, which is then completely explored (up to the desired precision) when box B1,1 is considered. With the same argumentation the boxes B1,4, B1,7 and

B1,5 are added after the position of B1,1. Then, the algorithm first explores box B1,2 completely, since also the boxes from the new partition are added at the beginning of the list S; and, thereafter, the algorithm goes over to explore box B1,3 and so on.

Assuming that the insertion in lines 23 and 26 in Algorithm 5.1 is implemented in such a way that the new element is always added to the beginning of the corresponding list

S, we change the for-loop in line 22 to

for i = 5, 7, 4, 1, 6, 3, 2 do

such that the above derived ordering is obtained. Moreover, with the same argumenta- tion, we substitute the for-loop in line 25 with

for i = 3, 2, 4 do .

Analogous arguments show that both substitutions are also valid if problem (P_ε2

1,ε3) or

(P_ε3_2,ε2) was called.

A remaining issue of the nondominated selection rule are the outcomes lying on the boundary of some box since such a point can belong to more than one box and, thus, such found outcomes can be dominated in a later iteration step. However, we assume here that such situations do not occur during the run of the algorithm. In practice, this issue can be handled numerically if some small predefined box-gap between two neighbored boxes is used (cf. Appendix A.2).

To conclude, using the nondominated selection rule, each found point cannot be dominated by a point from a later iteration step. However, a new point can still be dominated by a former found point. If the threshold for partial dominance is chosen sufficiently small, the modified algorithm returns a representative system for which only few or even no points in the set Rep are dominated by any other point in Rep. Hence, with appropriately chosen threshold, the nondominated selection rule is able to combine the coverage property with the quality guarantee from Corollary 5.17.