Progressively Interactive Evolutionary Multi-Objective Optimization ...

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1 Introduction Many real-world applications of multi-objective optimization involve a high number of objectives. Existing evolutionary multi-objective optimization algorithms [7, 34] have been applied to problems having multiple objectives for the task of finding a well-representative set of Paretooptimal solutions [6, 4]. These methods have been successful in solving a wide variety of problems with two or three objectives. However, these methodologies tend to fail for high number of objectives (greater than three) [8, 22]. The major hindrances in handling high number of objectives relate to stagnation in search, increased dimensionality of Pareto-optimal front, large computational cost, and difficulty in visualization of the objective space. These difficulties are inherent to a multi-objective problem having a high number of dimensions and cannot be eliminated; rather, procedures to handle such difficulties need to be explored. In many of the existing methodologies, preference information from the decision maker is utilized before the beginning of the search process or at the end of the search process to produce the optimal solution(s) in a multi-objective problem. Some approaches interact with the decision maker and iterate the process of elicitation and search until a satisfactory solution is found. However, not many studies have been performed where preference information is elicited during the search process and the information is utilized to progressively proceed towards the most preferred solution. This dissertation is an effort towards development of progressively interactive procedures to handle difficult multi-objective problems, combining concepts from the fields of Evolutionary Multi-objective Optimization (EMO) and Multi Criteria Decision Making (MCDM). The fields of Evolutionary Multi-objective Optimization and Multi Criteria Decision Making have a common goal, but researchers have shown only lukewarm interest, until recently, in applying the principles of one field to the other. In the dissertation, emphasis has been placed on integration of methods and development of hybrid procedures that are helpful in the extension of the existing algorithms to handle challenging problems with multiple objec- 3
tives. The utility of the procedures has also been shown on bilevel multiobjective optimization problems. The amalgamation of ideas has profound ramifications and addresses the challenges posed by multi-objective optimization problems. The dissertation is composed of five papers which have been summarized in this introductory chapter. Before providing a summary, a short review of the basic concepts necessary to understand the papers will be given in the following sections. 1.1 Multi-objective Optimization In a multi-objective optimization problem [30, 19, 6] there are two or more conflicting objectives which are supposed to be simultaneously optimized subject to a given set of constraints. These problems are commonly found in the fields of science, engineering, economics or any other field where optimal decisions are to be taken in the presence of trade-offs between two or more conflicting objectives. Usually such problems do not have a single solution which would simultaneously maximize/minimize each of the objectives; instead, there is a set of solutions which are optimal. These optimal solutions are called the Pareto-optimal solutions. A general multiobjective problem (M ≥ 2) can be described as follows: Maximize f(x) = (f1(x), . . . , fM(x)) , subject to g(x) ≥ 0, h(x) = 0, x (L) i ≤ xi ≤ x (U) i , i = 1, . . . , n. (1.1) In the above formulation, x represents the decision variable which lies in the decision space. The decision space is the search space represented by the constraints and variable bounds in a general multi-objective problem statement. The objective space f(x) is the image of the decision space under the objective function f. In a single objective optimization (M = 1) problem the feasible set is completely ordered according to the objective function f(x) = f1(x), such that for solutions, x (1) and x (2) in the decision space, either f1(x (1) ) ≥ f1(x (2) ) or f1(x (2) ) ≥ f1(x (1) ). Therefore, for two solutions in the objective space there are two possibilities with respect to the ≥ relation. However, when several objectives (M ≥ 2) are involved, the feasible set is not necessarily completely ordered, but partially ordered. In multiobjective problems, for any two objective vectors, f(x (1) ) and f(x (2) ), the relations =, > and ≥ can be extended as follows, • f(x (1) ) = f(x (2) ) fi(x (1) ) = fi(x (2) ) : ⇔ i ∈ {1, 2, . . . , M} 4
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 14 and 15: • f(x (1) ) ≥ f(x (2) ) fi(x (1
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
Page 45 and 46: mechanisms) and their emphasis of n
Page 47 and 48: f2 10 8 6 4 2 P1 P2 Most preferred
Page 49 and 50: DM calls. As mentioned earlier, the
Page 51 and 52: A linear value function similar to
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Page 61 and 62: TABLE III DISTANCE OF OBTAINED SOLU
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The PI-EMO-VF algorithm has been te
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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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f2 0000000000000 1111111111111 0000
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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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