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An Interactive Evolutionary Multi-Objective Optimization Method Based on Polyhedral Cones Ankur Sinha, Pekka Korhonen, Jyrki Wallenius and Kalyanmoy Deb ⋆⋆ Department of Business Technology Aalto University School of Economics PO Box 21220, 00076 Aalto, Helsinki, Finland {Firstname.Lastname}@hse.fi Abstract. This paper suggests a preference based methodology, where the information provided by the decision maker in the intermediate runs of an evolutionary multi-objective optimization algorithm is used to construct a polyhedral cone. This polyhedral cone is used to eliminate a part of the search space and conduct a more focussed search. The domination principle is modified, to look for better solutions lying in the region of interest. The search is terminated by using a local search based termination criterion. Results have been presented on two to five objective problems and the efficacy of the procedure has been tested. Keywords: Evolutionary multi-objective optimization, multiple criteria decisionmaking, interactive multi-objective optimization, sequential quadratic programming, preference based multi-objective optimization. 1 Introduction Most of the existing evolutionary multi-objective optimization (EMO) algorithms aim to find a set of well-converged and well-diversified Pareto-optimal solutions [1, 2]. As discussed elsewhere [3, 5], finding the entire set of Pareto-optimal solutions has its own intricacies. Firstly, the usual domination principle allows a majority of the population members to become non-dominated, thereby not allowing much room for introducing new solutions in a finite population. This slows down the progress of an EMO algorithm. Secondly, the representation of a high-dimensional Pareto-optimal front requires an exponentially large number of points, thereby requiring a large population size in running an EMO procedure. Thirdly, the visualization of a high-dimensional front becomes a non-trivial task for decision-making purposes. In most of the existing EMO algorithms the decision maker is usually not involved during the optimization process. The decision maker is called only at the end of the optimization run after a set of approximate Pareto-optimal solutions has been found. The decision making process is then executed by choosing the most preferred solution from the set of approximate Pareto-optimal solutions obtained. This approach is called ⋆⋆ Also Department of Mechanical Engineering, Indian Institute of Technology Kanpur, PIN 208016, India (deb@iitk.ac.in). 58
the aposteriori approach. This approach makes it necessary for the algorithm to produce the entire set of approximate Pareto-optimal solutions so that a decision making process can be executed. Some EMO researchers adopt a particular multiple criteria decision-making (MCDM) approach (apriori approach) to avoid the problems associated with finding the entire front. In such an approach the entire Pareto-optimal set is not aimed at rather a crowded set of Pareto-optimal solutions near the most preferred solution is targeted. In this approach the decision maker interacts at the beginning of an EMO run. The conedomination based EMO [6, 1], biased niching based EMO [7], reference point based EMO approaches [8, 9], the reference direction based EMO [10], the light beam approach based EMO [11] are a few attempts in this direction. In a semi-interactive EMO approach, the decision maker is involved iteratively [13, 14] in the optimization process. Some preference information (in terms of reference points or reference directions or others) is accepted from the decision maker and an MCDM-based EMO algorithm is employed to find a set of preferred Pareto-optimal solutions. Thereafter, a few representative preferred solutions are shown to the DM and a second set of preference information in terms of new reference points or new reference directions is obtained and a second MCDM-based EMO run is made. This procedure is continued till a satisfactory solution is found. However, the decision maker could be integrated with the optimization run of an EMO algorithm in a much more effective way, as shown in recent studies [4, 15]. These approaches require progressive interaction with the decision maker during the intermediate generations of the optimization process to converge towards the most preferred solution. Such a progressively interactive EMO approach (PI-EMO), allows the decision maker to modify her/his preference structure as new solutions evolve, thus making the process more DM-oriented. This paper discusses a simple PI-EMO where the decision maker is provided with a set of points perodically and asked to pick the most preferred solution from the set. Each time the decision maker is asked to make a choice of the most preferred solution, we call the instance as a ‘DM call’. With the information obtained from the decision maker a polyhedral cone is constructed and the domination principle is modified. The obtained polyhedral cone is further utilized to figure out a direction in which a local search is performed to determine the termination of the PI-EMO algorithm. The PI- EMO concept has been integrated with the NSGA-II algorithm [18] and the working of the algorithm has been demonstrated on three test problems having two, three and five objectives. A parametric study of the algorithm has also been done to determine the overall working of the algorithm. 2 Past Studies on Progressively Interactive Methods Towards the methodologies involving a progressive use of preference information by involving a decision-maker in an evolutionary multi-objective optimization framework, there are not many studies yet. Some recent studies periodically presented to the DM one or more pairs of alternative points found by an EMO algorithm and expected the DM to provide some preference information about the points. Some of the work in this 59
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Department of Business Technology P
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Aalto University publication series
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Progressively Interactive Evolution
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Acknowledgements The dissertation h
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II List of Papers 27 1 An interacti
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1 Introduction Many real-world appl
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• 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
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Page 24 and 25: Minimize/Maximize F(x) = (f1(x), f2
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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
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Page 41 and 42: In Step 2, points in the best non-d
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Page 47 and 48: f2 10 8 6 4 2 P1 P2 Most preferred
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Page 59 and 60: f 2 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 P
Page 61 and 62: TABLE III DISTANCE OF OBTAINED SOLU
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Page 74 and 75: Table 1. Final solutions obtained b
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Page 78 and 79: DM Calls and Func. Evals. (in thous
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Page 86 and 87: 3.2 AlgorithmicDevelopments Onesimp
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Page 92 and 93: F2 Step 3: Map (U1,U2) to (f1*,f2*)
Page 94 and 95: F2 1.4 1.2 1 0.8 0.6 0.4 0.2 0 A y1
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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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