Progressively Interactive Evolutionary Multi-Objective Optimization ...

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• f(x (1) ) ≥ f(x (2) ) fi(x (1) ) ≥ fi(x (2) ) : ⇔ i ∈ {1, 2, . . . , M} • f(x (1) ) > f(x (2) ) ⇔ f(x (1) ) ≥ f(x (2) ) ∧ f(x (1) ) = f(x (2) ) While comparing the multi-objective scenario with the single objective case [5], in contrast we find that for two solutions in the objective space there are three possibilities with respect to the ≥ relation. These possibilities are: f(x (1) ) ≥ f(x (2) ), f(x (2) ) ≥ f(x (1) ) or f(x (1) ) f(x (2) ) ∧ f(x (2) ) f(x (1) ). If any of the first two possibilities are met, it allows to rank or order the solutions independent of any preference information (or a decision maker). On the other hand, if the first two possibilities are not met, the solutions cannot be ranked or ordered without incorporating preference information (or involving a decision maker). Drawing analogy from the above discussion, the relations < and ≤ can be extended in a similar way. 1.2 Domination Concept and Optimality 1.2.1 Domination Concept Based on the established binary relations for two vectors in the previous section, the following domination concept [14] can be constituted, • x (1) strongly dominates x (2) ⇔ f(x (1) ) > f(x (2) ), • x (1) weakly dominates x (2) ⇔ f(x (1) ) ≥ f(x (2) ), • x (1) and x (2) are non-dominated with respect to each other⇔ f(x (1) ) f(x (2) ) ∧ f(x (2) ) f(x (1) ). The above domination concept is also explained in Figure 1.1 for a two objective maximization case. In Figure 1.1 two shaded regions have been shown in reference to point A. The shaded region in the north-east corner (excluding the lines) is the region which strongly dominates point A, the shaded region in the south-west corner (excluding the lines) is strongly dominated by point A and the unshaded region is the non-dominated region. Therefore, point A strongly dominates point B, points A, E and D are non-dominated with respect to each other, and point A weakly dominates point C. Most of the existing evolutionary multi-objective optimization algorithms use the domination principle to converge towards the optimal set of solutions. The concept allows us to order two decision vectors based on the corresponding objective vectors in the absence of any preference 5
f2 D C Region Dominated by A B A Region Dominating A Figure 1.1: Explanation for the domination concept for a maximization problem where A strongly dominates B; A weakly dominates C; A, D and E are non-dominated. E f1 f2 4 Non−dominated Set 1 2 3 4 2 5 Pareto−optimal Front indifference incomparable preference Figure 1.2: Explanation for the concept of a non-dominated set and a Pareto-optimal front. A hypothetical decision maker’s preferences for binary pairs are also shown. information. The algorithms which operate with a sparse set of solutions in the decision space and the corresponding images in the objective space usually give priority to a solution which dominates another solution. The solution which is not dominated with respect to any other solution in the sparse set is referred to as a non-dominated solution. In case of a discrete set of solutions: the subset whose solutions are not dominated by any solution in the discrete set is referred to as the non-dominated set within the discrete set. The non-dominated set consists of the best solutions available and form a front called a non-dominated front. When the set in consideration is the entire search space, the resulting non-dominated set is referred as a Pareto-optimal set and the front is referred as the Pareto-optimal front. To formally define a Pareto-optimal set, consider a set X, which constitutes the entire decision space with solutions x ∈ X. The subset X ∗ : X ∗ ⊂ X, containing solutions x ∗ , which are not dominated by any x in the entire decision space forms a Paretooptimal set. The concept of a Pareto-optimal front and a non-dominated front are illustrated in Figure 1.2. The shaded region in the figure represents f(x) : x ∈ X. It is the image in the objective space of the entire feasible region in the decision space. The bold curve represents the Pareto-optimal front 6 1 5 2 3 f1
Page 1 and 2: Department of Business Technology P
Page 3 and 4: Aalto University publication series
Page 5 and 6: Progressively Interactive Evolution
Page 7: Acknowledgements The dissertation h
Page 10 and 11: II List of Papers 27 1 An interacti
Page 12 and 13: 1 Introduction Many real-world appl
Page 16 and 17: for a maximization problem. Mathema
Page 18 and 19: • x (1) ∼ x (2) ⇔ x (1) and x
Page 20 and 21: Start Initialise Population Assign
Page 22 and 23: ithm can approximate the Pareto-opt
Page 24 and 25: Minimize/Maximize F(x) = (f1(x), f2
Page 26 and 27: the search is chosen. A scalarizing
Page 28 and 29: of the papers is provided in this s
Page 30: problems. The study discusses some
Page 33 and 34: [9] K. Deb and A. Sinha. An efficie
Page 35 and 36: [31] L. Thiele, K. Miettinen, P. Ko
Page 37 and 38: 1 An interactive evolutionary multi
Page 39 and 40: DM and an MCDM-based EMO algorithm
Page 41 and 42: In Step 2, points in the best non-d
Page 43 and 44: esulting constraints then become ki
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Page 47 and 48: f2 10 8 6 4 2 P1 P2 Most preferred
Page 49 and 50: DM calls. As mentioned earlier, the
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An Interactive Evolutionary Multi-O
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direction has been done by Phelps a
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0 ∀ i ∈ {1, . . . , M}, then th
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Table 1. Final solutions obtained b
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Table 4. Distance of obtained solut
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DM Calls and Func. Evals. (in thous
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13. P. Korhonen and J. Laakso, “A
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An EfficientandAccurateSolution Met
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eight differentproblems areshown. A
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3.2 AlgorithmicDevelopments Onesimp
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more studies are performed, the alg
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4.4 TestProblemTP4 The next problem
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F2 Step 3: Map (U1,U2) to (f1*,f2*)
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F2 1.4 1.2 1 0.8 0.6 0.4 0.2 0 A y1
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• Theupperlevelproblemhas multi-m
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are K + L + 1real-valuedvariablesin
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thereafterinaself-adaptivemannerbyd
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ulation of Nl(x (1) u ) lower level
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NSGA-II is able to bring the member
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F2 0 −0.2 −0.4 −0.6 −0.8
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LL Function Evals. UL Function Eval
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Table 1: Total function evaluations
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Difference in HV, DH(T) 350000 3000
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Table 4: Comparison of function eva
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Deb, K. and Sinha, A. (2009a). Cons
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Sun, D., Benekohal, R. F., and Wall
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Bilevel Multi-Objective Optimizatio
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The constraint functions g(x) and h
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value function which are required t
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to get close to the most preferred
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F2 0.4 0.2 0 −0.2 Lower level P
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F2 2 1.5 1 0.5 Most Preferred Point
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provement in terms of function eval
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9HSTFMG*aeafcd+ ISBN: 978-952-60-40
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