Covering Pareto Sets by Multilevel Subdivision Techniques

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 26f 1f 2540354302532015210510| |ab0010203040 0 5 10 15 20 25 30 35 40Figure 3: Examples of (MOP)s with optima relative to the boundary. Left: thepoint a is a Pareto point of the (MOP) given by F (x) =(f 1 (x),f 2 (x)) and Q =[a, b].Right: a covering of the set of local Pareto points. For a detailed discussion of thisparticular (MOP) we refer to Sec. 5.3.a local and a global Pareto point. Furthermore the efficiency of the algorithm willget worse when the (MOP) has optima relative to the boundary of the domain.The recovering algorithm is able to extend the computed box covering of the setof substationary points but it is just local in nature.The sampling algorithm is able to detect global Pareto points even on the boundaryof the domain due to the fact that it works in the image space of the (MOP).However, there remains always uncertainty because of the sampling approach, inparticular when the boxes are big and/or the dimension of the (MOP) is large. Neverthelessit turned out in practice that this algorithm works quite well, in particularwhen the gradients of the objectives are not available and the dimension of the(MOP) is moderate.To obtain an even better better performance – i.e. to compute a robust approximationof the Pareto set and to use a moderate amount of function calls – wepropose the following combination of the three algorithms. Here we assume that thegradients of all objectives are available.step 1 Start with the subdivision algorithm because of its robustness. Take a fewtest points for the evaluation of the boxes via the dynamical system P .step 2 Apply the recovering algorithm to the box collection which has been computedin step 1. This fills the gaps which have possibly been generated in step 1.step 3 Use the sampling algorithm to tighten the extended box collection. By usingthis algorithm boxes which only contain local Pareto points can be removedfrom the covering. Furthermore boxes get removed which contain no substa-17
tionary points but where kept in step 1 because of the weak convergence of Pin these regions.There are of course other possible ways to combine the algorithms, e.g. it ispossible to apply again the first algorithm on the box collection obtained by theprocedure described above. In step 2 the number of boxes which are added to thecollection is a measure for the number of test points needed in step 1.5 Numerical ResultsIn this section we illustrate the efficiency of our set oriented algorithms by severalnumerical examples.5.1 Example 1We begin by considering a simple example in order to illustrate the working principleof the subdivision procedure.The (MOP) is given by two objective functions f 1 ,f 2 : R 2 → R,f 1 (x 1 ,x 2 )=(x 1 − 1) 2 +(x 2 − 1) 4 ,f 2 (x 1 ,x 2 )=(x 1 +1) 2 +(x 2 +1) 2 .The basic region in parameter space is chosen to be Q =[−5, 5] × [−5, 5]. In Fig. 4we show box coverings generated by the subdivision algorithm after several stepsindicating that these sets indeed converge to the set of Pareto points.5.2 Example 2In this example we consider three objective functions f 1 ,f 2 ,f 3 : R 3 → R,f 1 (x 1 ,x 2 ,x 3 )=(x 1 − 1) 4 +(x 2 − 1) 2 +(x 3 − 1) 2 ,f 2 (x 1 ,x 2 ,x 3 )=(x 1 +1) 2 +(x 2 +1) 4 +(x 3 +1) 2 ,f 3 (x 1 ,x 2 ,x 3 )=(x 1 − 1) 2 +(x 2 +1) 2 +(x 3 − 1) 4 .The basic domain is chosen as Q =[−5, 5] 3 . Also in this higher dimensional casethe subdivision algorithm can successfully be applied. Resulting box collections areshown in Figs. 5and 6.18
Page 1 and 2: Covering Pareto Sets by MultilevelS
Page 3: entire Pareto set and not just sing
Page 6 and 7: Remark 2.6 Under certain smoothness
Page 8 and 9: Definition 2.9 Let g 1 ,...,g s :
Page 10 and 11: 3 Convergence towards Pareto SetsBy
Page 12 and 13: Corollary 3.2 Suppose that the set
Page 14 and 15: with the given point x j we evaluat
Page 16 and 17: (i) for all B ∈B kB.active := TRU
Page 20 and 21: 5948372610−1−2−3−4−5−5
Page 22 and 23: Figure 6: The box covering B 33 . T
Page 24 and 25: −11110.50.50.5000−0.5−0.5−0
Page 26 and 27: As we have already seen in Fig. 10
Page 28 and 29: 6 ConclusionIn this article we pres

Covering Pareto Sets by Multilevel Subdivision Techniques

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