Evolutionary many-objective optimization: A quick-start guide

Chand, Shelvin; Wagner, Markus

doi:10.1016/j.sorms.2015.08.001

Cited by 92 publications

(68 citation statements)

References 74 publications

(102 reference statements)

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“…We do this by ensuring that all solutions are incomparable when applying these indicators. For a more comprehensive introduction to dominance we refer the interested reader to [4], which is present in a large number of multi-objective optimization indicators.…”

Section: Indicator-based Diversity Optimizationmentioning

confidence: 99%

Evolutionary diversity optimization using multi-objective indicators

Neumann

Gao

Wagner

et al. 2019

Proceedings of the Genetic and Evolutionary Computation Conference

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Evolutionary diversity optimization aims to compute a diverse set of solutions where all solutions meet a given quality criterion. With this paper, we bridge the areas of evolutionary diversity optimization and evolutionary multi-objective optimization. We show how popular indicators frequently used in the area of multi-objective optimization can be used for evolutionary diversity optimization. Our experimental investigations for evolving diverse sets of TSP instances and images according to various features show that two of the most prominent multi-objective indicators, namely the hypervolume indicator and the inverted generational distance, provide excellent results in terms of visualization and various diversity indicators.

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Section: Indicator-based Diversity Optimizationmentioning

confidence: 99%

Evolutionary diversity optimization using multi-objective indicators

Neumann

Gao

Wagner

et al. 2019

Proceedings of the Genetic and Evolutionary Computation Conference

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“…According to Pareto ordering, i.e., In industrial applications, obtaining the whole Pareto set/front rather than a single solution enables us to compare promising alternatives and to explore new innovative designs, whose concept is variously refered to as innovization (Deb, Sinha, and Kukkonen 2006), multi-objective design exploration (Obayashi, Jeong, and Chiba 2005) and design informatics (Chiba, Makino, and Takatoya 2009). Quite a few real-world problems involve simulations and/or experiments to evaluate solutions (Chand and Wagner 2015) and lack the mathematical expression of their objective functions and derivatives. Multi-objective evolutionary algorithms are a tool to solve such problems where the Pareto set/front is approximated by a population, i.e., a finite set of sample points (Coello, Lamont, and Van Veldhuisen 2007).…”

Section: Introductionmentioning

confidence: 99%

Bézier Simplex Fitting: Describing Pareto Fronts of´ Simplicial Problems with Small Samples in Multi-Objective Optimization

Kobayashi

Hamada

Sannai

et al. 2019

AAAI

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Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an M -objective optimization problem often forms an (M − 1)-dimensional topological simplex (a curved line for M = 2, a curved triangle for M = 3, a curved tetrahedron for M = 4, etc.). Since the dimensionality of the solution set increases as the number of objectives grows, an exponentially large sample size is needed to cover the solution set. To reduce the required sample size, this paper proposes a Bézier simplex model and its fitting algorithm. These techniques can exploit the simplex structure of the solution set and decompose a high-dimensional surface fitting task into a sequence of low-dimensional ones. An approximation theorem of Bézier simplices is proven. Numerical experiments with synthetic and real-world optimization problems demonstrate that the proposed method achieves an accurate approximation of high-dimensional solution sets with small samples. In practice, such an approximation will be conducted in the postoptimization process and enable a better trade-off analysis. PreliminariesLet us introduce notations for defining simplicial problems and review an existing method of Bézier curve fitting. Simplicial ProblemA multi-objective optimization problem is denoted by its objective map f = (f 1 , . . . , f M ) : X → R M . Let I := { 1, . . . , M } be the index set of objective functions and ∆ M −1 := (t 1 , . . . , t M ) ∈ R M 0 ≤ t m , m∈M t m = 1 be the standard simplex in R M . For each non-empty subset J ⊆ I, we call∆ J := (t 1 , . . . , t M ) ∈ ∆ M −1 t m = 0 (m ∈ J) the J-face of ∆ M −1 andThe problem class we are interested in is as follows:We call such φ and f • φ a triangulation of the Pareto set X * (f ) and the Pareto front f X * (f ), respectively. For each non-empty subset J ⊆ I, we call X * (f J ) the J-face of X * (f ) and f X * (f J ) the J-face of f X * (f ). For each 0 ≤ m ≤ M − 1, we callthe m-skeleton of X * (f ) and f X * (f ), respectively.By definition, any subproblem of a simplicial problem is again simplicial. As shown in Figure 1b, the Pareto sets forms a simplex. The second condition asserts that f | X * (f ) : X * (f ) → R M is a C 0 -embedding. This means that the Pareto front of each subproblem is homeomorphic to its Pareto set as shown in Figure 1c. Therefore, the Pareto set/front of an M -objective simplicial problem can be identified with a curved (M − 1)-simplex. We can find its J-face by solving the J-subproblem.

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“…However, in the Target Problems , the number of objectives cannot be always reduced to two or three. In cases such as this one, graphical visualization techniques of nondominated solutions in the objective space with larger than three objectives play an important role in communicating with the decision maker (Lücken et al, ; Chand & Wanger, ).…”

Section: Introductionmentioning

confidence: 99%

A many‐objective evolutionary algorithm incorporating decision maker's preference and its application to management of the electricity distribution network

Sekizaki

Nishizaki

Hayashida

2019

Multi Criteria Decision Anal

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This paper proposes a step‐by‐step decision making approach that incorporates preferences of a decision maker into a many‐objective evolutionary algorithm. The algorithm is applied to a challenging many‐objective optimization problem (MaOP) involving integer objective values, tight constraints, heavy computational burden, and limited knowledge about the true Pareto front. The proposed method consists of five steps. (a) The entire Pareto front with high dimensions is roughly approximated by a small set of nondominated solutions. (b) On the basis of the approximated Pareto front, the decision maker specifies a preference. (c) Reference points are distributed through part of the objective space using the preference information. (d) The approximation accuracy of the Pareto front in the region of interest is improved by local search using the reference points. (e) The most preferred solution from a set of the nondominated solutions is chosen. The evolutionary algorithms employed in Steps (a) and (d) have an improved normalization procedure and selection mechanism for the MaOP involving integer objective value(s) and the tight constraints. Steps (b) and (e) use a graphical‐user‐interface‐based interactive tool we developed to efficiently narrow down to the preferred solution(s) among the high‐dimensional, nondominated solutions. This alleviates the cognitive burden on the decision maker and therefore aids the decision maker in handling MaOPs. In the computational experiment, we choose the real‐world application of reconfiguring an electricity distribution network. We demonstrate that the proposed approach can provide satisfying solutions to a decision maker.

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Evolutionary many-objective optimization: A quick-start guide

Cited by 92 publications

References 74 publications

Evolutionary diversity optimization using multi-objective indicators

Evolutionary diversity optimization using multi-objective indicators

Bézier Simplex Fitting: Describing Pareto Fronts of´ Simplicial Problems with Small Samples in Multi-Objective Optimization

A many‐objective evolutionary algorithm incorporating decision maker's preference and its application to management of the electricity distribution network

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