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Olivier Ladislas De Weck - One of the best experts on this subject based on the ideXlab platform.
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Adaptive weighted sum method for multiobjective optimization: a new method for Pareto Front generation
Structural and Multidisciplinary Optimization, 2006Co-Authors: Olivier Ladislas De WeckAbstract:This paper presents an adaptive weighted sum (AWS) method for multiobjective optimization problems. The method extends the previously developed biobjective AWS method to problems with more than two objective functions. In the first phase, the usual weighted sum method is performed to approximate the Pareto surface quickly, and a mesh of Pareto Front patches is identified. Each Pareto Front patch is then refined by imposing additional equality constraints that connect the pseudonadir point and the expected Pareto optimal solutions on a piecewise planar hypersurface in the $$ {m} $$ -dimensional objective space. It is demonstrated that the method produces a well-distributed Pareto Front mesh for effective visualization, and that it finds solutions in nonconvex regions. Two numerical examples and a simple structural optimization problem are solved as case studies.
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Adaptive weighted-sum method for bi-objective optimization: Pareto Front generation
Structural and Multidisciplinary Optimization, 2005Co-Authors: I. Y. Kim, Olivier Ladislas De WeckAbstract:This paper presents a new method that effectively determines a Pareto Front for bi-objective optimization with potential application to multiple objectives. A traditional method for multiobjective optimization is the weighted-sum method, which seeks Pareto optimal solutions one by one by systematically changing the weights among the objective functions. Previous research has shown that this method often produces poorly distributed solutions along a Pareto Front, and that it does not find Pareto optimal solutions in non-convex regions. The proposed adaptive weighted sum method focuses on unexplored regions by changing the weights adaptively rather than by using a priori weight selections and by specifying additional inequality constraints. It is demonstrated that the adaptive weighted sum method produces well-distributed solutions, finds Pareto optimal solutions in non-convex regions, and neglects non-Pareto optimal solutions. This last point can be a potential liability of Normal Boundary Intersection, an otherwise successful multiobjective method, which is mainly caused by its reliance on equality constraints. The promise of this robust algorithm is demonstrated with two numerical examples and a simple structural optimization problem.
Miqing Li - One of the best experts on this subject based on the ideXlab platform.
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what weights work for you adapting weights for any Pareto Front shape in decomposition based evolutionary multiobjective optimisation
Evolutionary Computation, 2020Co-Authors: Miqing LiAbstract:The quality of solution sets generated by decomposition-based evolutionary multi-objective optimisation (EMO) algorithms depends heavily on the consistency between a given problem's Pareto Front sh...
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what weights work for you adapting weights for any Pareto Front shape in decomposition based evolutionary multi objective optimisation
arXiv: Neural and Evolutionary Computing, 2017Co-Authors: Miqing LiAbstract:The quality of solution sets generated by decomposition-based evolutionary multiobjective optimisation (EMO) algorithms depends heavily on the consistency between a given problem's Pareto Front shape and the specified weights' distribution. A set of weights distributed uniformly in a simplex often lead to a set of well-distributed solutions on a Pareto Front with a simplex-like shape, but may fail on other Pareto Front shapes. It is an open problem on how to specify a set of appropriate weights without the information of the problem's Pareto Front beforehand. In this paper, we propose an approach to adapt the weights during the evolutionary process (called AdaW). AdaW progressively seeks a suitable distribution of weights for the given problem by elaborating five parts in the weight adaptation --- weight generation, weight addition, weight deletion, archive maintenance, and weight update frequency. Experimental results have shown the effectiveness of the proposed approach. AdaW works well for Pareto Fronts with very different shapes: 1) the simplex-like, 2) the inverted simplex-like, 3) the highly nonlinear, 4) the disconnect, 5) the degenerated, 6) the badly-scaled, and 7) the high-dimensional.
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a performance comparison indicator for Pareto Front approximations in many objective optimization
Genetic and Evolutionary Computation Conference, 2015Co-Authors: Miqing Li, Shengxiang YangAbstract:Increasing interest in simultaneously optimizing many objectives (typically more than three objectives) of problems leads to the emergence of various many-objective algorithms in the evolutionary multi-objective optimization field. However, in contrast to the development of algorithm design, how to assess many-objective algorithms has received scant concern. Many performance indicators are designed in principle for any number of objectives, but in practice are invalid or infeasible to be used in many-objective optimization. In this paper, we explain the difficulties that popular performance indicators face and propose a performance comparison indicator (PCI) to assess Pareto Front approximations obtained by many-objective algorithms. PCI evaluates the quality of approximation sets with the aid of a reference set constructed by themselves. The points in the reference set are divided into many clusters, and the proposed indicator estimates the minimum moves of solutions in the approximation sets to weakly dominate these clusters. PCI has been verified both by an analytic comparison with several well-known indicators and by an empirical test on four groups of Pareto Front approximations with different numbers of objectives and problem characteristics.
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diversity comparison of Pareto Front approximations in many objective optimization
IEEE Transactions on Systems Man and Cybernetics, 2014Co-Authors: Miqing Li, Shengxiang YangAbstract:Diversity assessment of Pareto Front approximations is an important issue in the stochastic multiobjective optimization community. Most of the diversity indicators in the literature were designed to work for any number of objectives of Pareto Front approximations in principle, but in practice many of these indicators are infeasible or not workable when the number of objectives is large. In this paper, we propose a diversity comparison indicator (DCI) to assess the diversity of Pareto Front approximations in many-objective optimization. DCI evaluates relative quality of different Pareto Front approximations rather than provides an absolute measure of distribution for a single approximation. In DCI, all the concerned approximations are put into a grid environment so that there are some hyperboxes containing one or more solutions. The proposed indicator only considers the contribution of different approximations to nonempty hyperboxes. Therefore, the computational cost does not increase exponentially with the number of objectives. In fact, the implementation of DCI is of quadratic time complexity, which is fully independent of the number of divisions used in grid. Systematic experiments are conducted using three groups of artificial Pareto Front approximations and seven groups of real Pareto Front approximations with different numbers of objectives to verify the effectiveness of DCI. Moreover, a comparison with two diversity indicators used widely in many-objective optimization is made analytically and empirically. Finally, a parametric investigation reveals interesting insights of the division number in grid and also offers some suggested settings to the users with different preferences.
Hisao Ishibuchi - One of the best experts on this subject based on the ideXlab platform.
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CEC - Regular Pareto Front Shape is not Realistic
2019 IEEE Congress on Evolutionary Computation (CEC), 2019Co-Authors: Hisao Ishibuchi, Linjun He, Ke ShangAbstract:Performance of evolutionary multi-objective and many-objective optimization algorithms is usually evaluated by computational experiments on a number of test problems. Thus, performance comparison results depend on the choice of test problems. For fair comparison, it is needed to use a wide variety of test problems with various characteristics. However, most of well-known and frequently-used scalable test problems have the same type of Pareto Fronts called "regular" Pareto Fronts: Their shape is triangular. In this paper, we discuss the reality of this type of Pareto Fronts. First, we show that a triangular Pareto Front has some unrealistic properties as the Pareto Front of a real-world multi-objective problem. Next, we examine the shape of the Pareto Fronts of some other multi-objective test problems with independently generated objectives (i.e., with objectives that are not derived from a pre-specified shape of Pareto Fronts). It is shown that the Pareto Fronts of those test problems are inverted triangular (i.e., not regular). Then, we demonstrate that the shape of Pareto Fronts (i.e., triangular or inverted triangular) has large effects on the performance of decomposition-based and hypervolume-based algorithms. Finally, we show difficulties of hypervolume-based performance evaluation for many-objective problems with inverted triangular Pareto Fronts.
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regular Pareto Front shape is not realistic
Congress on Evolutionary Computation, 2019Co-Authors: Hisao Ishibuchi, Linjun He, Ke ShangAbstract:Performance of evolutionary multi-objective and many-objective optimization algorithms is usually evaluated by computational experiments on a number of test problems. Thus, performance comparison results depend on the choice of test problems. For fair comparison, it is needed to use a wide variety of test problems with various characteristics. However, most of well-known and frequently-used scalable test problems have the same type of Pareto Fronts called "regular" Pareto Fronts: Their shape is triangular. In this paper, we discuss the reality of this type of Pareto Fronts. First, we show that a triangular Pareto Front has some unrealistic properties as the Pareto Front of a real-world multi-objective problem. Next, we examine the shape of the Pareto Fronts of some other multi-objective test problems with independently generated objectives (i.e., with objectives that are not derived from a pre-specified shape of Pareto Fronts). It is shown that the Pareto Fronts of those test problems are inverted triangular (i.e., not regular). Then, we demonstrate that the shape of Pareto Fronts (i.e., triangular or inverted triangular) has large effects on the performance of decomposition-based and hypervolume-based algorithms. Finally, we show difficulties of hypervolume-based performance evaluation for many-objective problems with inverted triangular Pareto Fronts.
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reference point specification in inverted generational distance for triangular linear Pareto Front
IEEE Transactions on Evolutionary Computation, 2018Co-Authors: Hisao Ishibuchi, Ryo Imada, Yu Setoguchi, Yusuke NojimaAbstract:The hypervolume and the inverted generational distance (IGD) have been frequently used for the comparison of evolutionary multiobjective optimization algorithms. For the hypervolume, the relation between the location of a reference point and the optimal distribution of solutions has been studied in the literature. However, such a relation has not been studied for the IGD whereas IGD-based comparison results depend on the specification of reference points. Our intention is to clearly demonstrate the dependency of IGD-based comparison results on reference point specification. First, we explain difficulties of fair comparison in the following two cases: one is the use of all nondominated solutions among obtained solutions by compared algorithms as reference points, and the other is the use of a small number of uniformly sampled reference points. Discussions on these two cases show the necessity of a large number of uniformly sampled reference points on the entire Pareto Front. Then, we show a bias of the IGD with such a reference point set through computational experiments. It is shown that the IGD tends to favor a solution set with much smaller diversity than a fully expanded solution set over the entire Pareto Front. Finally, we propose a new specification method where reference points are uniformly sampled not only from the Pareto Front but also from outside the Pareto Front.
Ke Shang - One of the best experts on this subject based on the ideXlab platform.
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regular Pareto Front shape is not realistic
Congress on Evolutionary Computation, 2019Co-Authors: Hisao Ishibuchi, Linjun He, Ke ShangAbstract:Performance of evolutionary multi-objective and many-objective optimization algorithms is usually evaluated by computational experiments on a number of test problems. Thus, performance comparison results depend on the choice of test problems. For fair comparison, it is needed to use a wide variety of test problems with various characteristics. However, most of well-known and frequently-used scalable test problems have the same type of Pareto Fronts called "regular" Pareto Fronts: Their shape is triangular. In this paper, we discuss the reality of this type of Pareto Fronts. First, we show that a triangular Pareto Front has some unrealistic properties as the Pareto Front of a real-world multi-objective problem. Next, we examine the shape of the Pareto Fronts of some other multi-objective test problems with independently generated objectives (i.e., with objectives that are not derived from a pre-specified shape of Pareto Fronts). It is shown that the Pareto Fronts of those test problems are inverted triangular (i.e., not regular). Then, we demonstrate that the shape of Pareto Fronts (i.e., triangular or inverted triangular) has large effects on the performance of decomposition-based and hypervolume-based algorithms. Finally, we show difficulties of hypervolume-based performance evaluation for many-objective problems with inverted triangular Pareto Fronts.
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CEC - Regular Pareto Front Shape is not Realistic
2019 IEEE Congress on Evolutionary Computation (CEC), 2019Co-Authors: Hisao Ishibuchi, Linjun He, Ke ShangAbstract:Performance of evolutionary multi-objective and many-objective optimization algorithms is usually evaluated by computational experiments on a number of test problems. Thus, performance comparison results depend on the choice of test problems. For fair comparison, it is needed to use a wide variety of test problems with various characteristics. However, most of well-known and frequently-used scalable test problems have the same type of Pareto Fronts called "regular" Pareto Fronts: Their shape is triangular. In this paper, we discuss the reality of this type of Pareto Fronts. First, we show that a triangular Pareto Front has some unrealistic properties as the Pareto Front of a real-world multi-objective problem. Next, we examine the shape of the Pareto Fronts of some other multi-objective test problems with independently generated objectives (i.e., with objectives that are not derived from a pre-specified shape of Pareto Fronts). It is shown that the Pareto Fronts of those test problems are inverted triangular (i.e., not regular). Then, we demonstrate that the shape of Pareto Fronts (i.e., triangular or inverted triangular) has large effects on the performance of decomposition-based and hypervolume-based algorithms. Finally, we show difficulties of hypervolume-based performance evaluation for many-objective problems with inverted triangular Pareto Fronts.
I Kim - One of the best experts on this subject based on the ideXlab platform.
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Adaptive weighted sum method for multiobjective optimization: a new method for Pareto Front generation
Structural and Multidisciplinary Optimization, 2006Co-Authors: I KimAbstract:Abstract This paper presents an adaptive weighted sum (AWS) method for multiobjective optimization problems. The method extends the previously developed biobjective AWS method to problems with more than two objective functions. In the first phase, the usual weighted sum method is performed to approximate the Pareto surface quickly, and a mesh of Pareto Front patches is identified. Each Pareto Front patch is then refined by imposing additional equality constraints that connect the pseudonadir point and the expected Pareto optimal solutions on a piecewise planar hypersurface in the
