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However, the ``symbol system'' refers to symbolic AI, and not a symbol system in our society. We humans still uses symbols, e.g., language, to think and communicate. Learning and using symbols is still a challenge in AI studies. To go further, we needed to disentangle the two different notions of symbols: symbol system... |
Clearly, studying the interaction between language and internal representations in the context of the more fluent, dynamic, and possibly more brain-like PSS is not out of date. We also believe that symbol emergence is not an auxiliary problem in cognitive developmental science, including cognitive developmental robotic... |
Tadahiro Taniguchi received the ME and PhD degrees from Kyoto University in 2003 and 2006, respectively. From April 2005 to March 2006, he was a Japan Society for the Promotion of Science (JSPS) research fellow (DC2) in the Department of Mechanical Engineering and Science, Graduate School of Engineering, Kyoto Universi... |
Emre Ugur is an Assistant Professor in Department of Computer Engineering Department, Bogazici University, Turkey. After receiving his PhD in Computer Engineering from Middle East Technical University, he worked at ATR Japan as a researcher (2009–2013), at University of Innsbruck as a senior researcher (2013–2016), and... |
Matej Hoffmann received the Mgr. (MSc) degree in computer science, artificial intelligence at Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic, in 2006, and the PhD degree from the Artificial Intelligence Laboratory, University of Zurich, Zurich, Switzerland, under the supervisio... |
He then served as a Senior Research Associate with the Artificial Intelligence Laboratory, University of Zurich, from 2012 to 2013. From 2013 to 2017, he was with the iCub Facility Department, Istituto Italiano di Tecnologia (IIT), Genoa, Italy, where he was a Marie Curie Experienced Researcher Fellow, from 2014 to 201... |
Lorenzo Jamone received his MS in Computer Engineering from the University of Genova in 2006 (with honors), and his PhD in Humanoid Technologies from the University of Genova and the IIT in 2010. He was a Research Fellow at the RBCS Department of the IIT in 2010, Associate Researcher at Takanishi Lab (Waseda University... |
Takayuki Nagai received his BE, ME, and PhD degrees from the Department of Electrical Engineering, Keio University, in 1993, 1995, and 1997, respectively. Since 1998, he has been with the University of Electro-Communications, where he is currently a professor of the Graduate School of Informatics and Engineering. From ... |
Benjamin Rosman is a Principal Researcher in the Mobile Intelligent Autonomous Systems group at the Council for Scientific and Industrial Research (CSIR) in South Africa, and is also a Senior Lecturer in the School of Computer Science and Applied Mathematics at the University of the Witwatersrand, where he runs the Rob... |
Toshihiko Matsuka received his PhD in Psychometrics from Columbia University in 2002. After receiving his PhD, he joined RUMBA lab at Rutgers University, Newark as a postdoctoral research fellow. While at Rutgers, he conducted research on cognitive neuroscience. He then moved to Stevens Institute of Technology as an As... |
Naoto Iwahashi received the BE degree in Engineering from Keio University, Yokohama, Japan, in 1985. He received the PhD degree in Engineering from Tokyo Institute of Technology, in 2001. In April 1985, he joined Sony Corp., Tokyo, Japan. From October 1990 to September 1993, he was at Advanced Telecommunications Resear... |
Erhan Oztop earned his PhD at the University of Southern California in 2002. In the same year, he joined the Computational Neuroscience Laboratories at the Advanced Telecommunications Research Institute International (ATR), in Japan. There, he worked as a researcher and later a senior researcher and group leader, in ad... |
Justus Piater is a professor of computer science at the University of Innsbruck, Austria, where he leads the Intelligent and Interactive Systems group. He holds a MSc degree from the University of Magdeburg, Germany, and MSc and PhD degrees from the University of Massachusetts Amherst, USA, all in computer science. Bef... |
Florentin Wörgötter has studied Biology and Mathematics in Düsseldorf. He received his PhD in 1988 in Essen, working experimentally on the visual cortex before he turned to computational issues at Caltech, USA (1988–1990). After 1990, he was a researcher at the University of Bochum, concerned with experimental and comp... |
# Source: arxiv 1802.08792 # Title: IGD Indicator-based Evolutionary Algorithm for Many-objective Optimization Problems # Sections: all # Downloaded: 2026-03-03T01:57:00.484172+00:00 |
IGD Indicator-based Evolutionary Algorithm for Many-objective Optimization Problems Abstract Inverted Generational Distance (IGD) has been widely considered as a reliable performance indicator to concurrently quantify the convergence and diversity of multi- and many-objective evolutionary algorithms. In this paper, an ... |
Index Terms: Nadir point, inverted generational distance, linear assignment problem, many-objective evolutionary optimization algorithm. |
Introduction Many-objective Optimization Problems (MaOPs) refer to the optimization tasks involving [MATH] (i.e., [MATH] ) conflicting objectives to be optimized concurrently |
. Generally, an MaOP is with the mathematic form represented by ( [EQUATION] where [MATH] is the feasible search space for the decision variables [MATH] , and [MATH] is the corresponding objective vector including [MATH] objectives which maps the [MATH] -dimensional decision space [MATH] to the [MATH] -dimensional obje... |
Optimization algorithms for addressing an MaOP aim at searching for a set of uniformly distributed solutions which are closely approximating the PF. Because the MaOPs widely exist in diverse real-world applications, such as policy management in land exploitation with [MATH] -objective |
and calibration of automotive engine with [MATH] -objective , to name a few, various algorithms for solving MaOPs have been developed. Among these algorithms, the evolutionary paradigms are considerably preferable due to their population-based meta-heuristic characteristics obtaining a set of quality solutions in a sin... |
During the past decades, various Multi-Objective Evolutionary Algorithms (MOEAs), such as elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) |
, advanced version of Strength Pareto Evolutionary Algorithm (SPEA2) , among others, have been proposed to effectively solve Multi-Objective Optimization Problems (MOPs). Unfortunately, these MOEAs do not scale well with the increasing number of objectives, mainly due to the loss of selection pressure. To be specific, ... |
caused by the curse of dimensionality , so that the traditional elitism mechanism based on Pareto-domination cannot effectively differentiate which solutions should survive into the next generation. As a result, the density-based diversity promotion mechanism is considered the sole mechanism for mating and environmenta... |
. However, the solutions with good diversity in MaOPs are generally not only distant from each other but also away from the PF. Consequently, the evolution with the solutions |
. To this end, various Many-Objective Evolutionary Algorithms (MaOEAs) specifically designed for addressing MaOPs have been proposed in recent years. |
Generally, these MaOEAs can be divided into four different categories. The first category covers the algorithms employing reference prior to enhancing the diversity promotion which in turn improve the convergence. For example, the MaOEA using reference-point-based non-dominated sorting approach (NSGA-III) |
employs a set of reference vectors to assist the algorithm to select solutions which are close to these reference vectors. Yuan et al. |
proposed the reference line-based algorithm which not only adopted the diversity improvement mechanism like that in NSGA-III but also introduced convergence enhancement scheme by measuring the distance between the origin to the solution projections on the corresponding reference line. In addition, a reference line-base... |
for explicitly promoting the diversity of an MaOEA. Furthermore, an approach (RVEA) was presented in to adaptively revise the reference vector positions based on the scales of the objective functions to balance the diversity and convergence. |
The second category refers to the decomposition-based algorithms which decompose an MaOP into several single-objective optimization problems, such as the MOEA based on Decomposition (MOEA/D) |
which was initially proposed for solving MOPs but scaled well for MaOPs. Specifically, MOEA/D transformed the original MOP/MaOP with [MATH] objectives into a group of single-objective optimization problems, and each sub-problem was solved in its neighboring region which constrained by their corresponding reference vect... |
of MOEA/D were proposed for improving the performance much further. The third category is known as the convergence enhancement-based approaches. More specifically, the traditional Pareto dominance comparison methods widely utilized in MOEAs are not effective in discriminating populations with good proximity in MaOPs. A... |
employed a relaxed factor [MATH] to compare the dominance relation between solutions; Pierro et al proposed the preference order ranking approach to replace the traditional non-dominated sorting. Furthermore, the fuzzy dominance methods |
studied the fuzzification of the Pareto-dominance relation to design the ranking scheme to select promising solutions; the [MATH] -optimality paradigm was proposed in |
to pick up solutions whose objectives were with the same importance by considering their objective value improvements. In addition, Yang et al. |
proposed the grid-based approach to select the solutions that have the higher priority of dominance, and control the proportion of Pareto-optimal solutions by adjusting the grid size. Meanwhile, Antonio et al. |
alternated the achievement function and the [MATH] -indicator method to improve the performance of MOEA in solving MaOPs. In , a modification of density estimation, termed as shift-based density estimation, was proposed to make the dominance comparison better suited for solving MaOPs. Furthermore, the favorable converg... |
to improve the selection pressure in mating and environmental selections. Recently, a knee point-based algorithm (KnEA) was presented as a secondary selection scheme to enhance the selection pressure. In summary, these algorithms introduced new comparison methods, designed effective selection mechanisms, or relaxed the... |
The fourth category is known as the indicator-based methods. For instance, several MOEAs based on the hypervolume (HV) were proposed in |
, however their major disadvantages were the costly overhead in calculating the HV values especially in solving MaOPs. To this end, Bader and Zitzler proposed the HypE method with the Monte Carlo simulation |
to estimate the HV value. Consequently, the computational cost was largely lowered compared to its predecessors whose HV values were calculated exactly. In |
, an [MATH] indicator-based algorithm ( [MATH] -EMOA) was proposed for solving bi-objective optimization problems, and then extended further for tri-objective problems |
. Furthermore, Villalobos and Coello integrated the [MATH] indicator with the differential evolution to solve MaOPs with up to [MATH] objectives. Recently, an Inverse Generational Distance Plus (IGD |
indicator-based evolutionary algorithm (IGD -EMOA) was proposed in for addressing MaOPs with no more than [MATH] objectives. Basically, the IGD indicator is viewed as a variant of the Inverse Generational Distance (IGD) indicator. |
Although the MaOEAs mentioned above have experimentally demonstrated their promising performance, major issues are easily to be identified in solving real-world applications. For example, it is difficult to choose the converting strategy of the MaOEAs from the second category, which motivates multiple variants |
to be developed further. In addition, the MaOEAs from the first and third categories only highlighted one of the characters in their designs (i.e., only the diversity promotion is explicitly concerned in the MaOEAs from the first category, and the convergence from the third category). However, both the diversity and co... |
, while the calculation of IGD is scalable without these deficiencies. In this paper, an IGD indicator-based Many-Objective Evolutionary Algorithm (MaOEA/IGD) has been proposed for effectively addressing MaOPs, and the contributions are outlined as follows: |
1. A Decomposition-based Nadir Point Estimation method (DNPE) has been presented to estimate the nadir points to facilitate the calculation of IGD indicator. In DNPE, the estimation focuses only on the extreme point areas and transforms the computation of an [MATH] -objective optimization problem into [MATH] single-obj... |
2. A comparison scheme for the non-dominated sorting has been designed for improving the convergence of the proposed algorithm. In this scheme, the dominance relations of solutions are not obtained by the comparisons among all the solutions but the solutions to the reference points. Therefore, the computational complex... |
3. Three types of proximity distance assignment mechanisms are proposed for the solutions according to their PF rank values, which make the solutions with good convergence in the same PF to have higher chances to be selected. Furthermore, these assignment mechanisms collectively assure the proposed IGD indicator to be ... |
4. Based on the proposed dominance comparison scheme and the proximity distance assignments, the selection mechanism which is employed for the mating selection and the environmental selection is proposed to concurrently facilitate the convergence and the diversity. |
The reminder of this paper is organized as follows. First, related works are reviewed, and the motivation of the proposed DNPE is presented in Section II . Then the details of the proposed algorithm are documented in Section III . To evaluate the performance of the proposed algorithm in addressing MaOPs, a series of ex... |
II Related Works and Motivation Literatures related to the nadir point estimation and IGD indicator-based EAs are thoroughly reviewed in this section. Specifically, the Worst Crowded NSGA-II (WC-NSGA-II) |
and the Pareto Corner Search Evolutionary Algorithm (PCSEA) would be reviewed and criticized in detail, because the insightful observations of the deficiencies of these two approaches naturally lead to the motivation of the proposed DNPE design. In addition, the IGD -EMOA is reviewed as well to highlight the utilizatio... |
II-A Nadir Point Estimation Methods According to literatures , the approaches for estimating the nadir points can be divided into three categories including the surface-to-nadir, edge-to-nadir, and extreme-point-to-nadir schemes. In the surface-to-nadir scheme, the nadir points are constructed from the current Pareto-o... |
and MaOEAs in belong to this category. However, these MOEAs are shown to perform poorly in MaOPs due to the curse of dimensionality |
. In addition, the MaOEAs related methods are not suitable for the proposed algorithm because the MaOPs have been solved prior to the nadir point estimation, while the nadir points in this paper are targeted for addressing MaOPs. |
The edge-to-nadir scheme covers the Marcin and Andrzej’s approach , Extremized Crowded NSGA-II (EC-NSGA-II) , and the recently proposed Emphasized Critical Region (ECR) approach |
. Specifically, Marcin and Andrzej’s approach decomposed an [MATH] -objective problem into [MATH] sub-problems to estimate the nadir point from the [MATH] edges, in which the major issues were the poor quality in nadir point found and the impractical computation complexity beyond three objectives |
. EC-NSGA-II modified the crowding distance of NSGA-II by assigning large rank values to the solutions which had the minimum or maximum objective values. The ECR emphasized the solutions lying at the edges of the PF (i.e., the critical regions) with the adopted MOEAs. Although EC-NSGA-II and ECR have been reported to b... |
The extreme-point-to-nadir approaches refer to employ a direct means to estimate the extreme points based on which the nadir points are derived, such as the Worst Crowded NSGA-II (WC-NSGA-II) |
in which the worst crowded solutions (extreme points) were preferred by ranking their crowding distances with large values. In WC-NSGA-II, it was hopeful that the extreme points were obtained when the evolution terminated. However, emphasizing the extreme points easily led to the WC-NSGA-II losing the diversity which i... |
to look for the nadir points with the corner-sort ranking method for the MaOPs whose objective values were required to be with the identical scales. |
In addition, there are also various methods not falling into the above categories. For example, Benayoun et al. estimated the nadir points with the pay-table |
in which the [MATH] -th row denoted the objective values of the solution which had the minima on its [MATH] -th objective. In addition, other related works were suggested in |
for the problems assuming a linear relationship between the objectives and variables. On the contrary, most of the real-world applications are non-linear in nature. |
Because the nadir point estimation is a critical part of the proposed algorithm for solving MaOPs, an approach with a high computational complexity is certainly not preferable. Furthermore, the nadir points are employed for constructing the Utopian PF, while the reference point of IGD would come from the PF. As a conse... |
Specifically, because the proposed nadir point estimation method (i.e., the DNPE) is based on the extreme-point-to-nadir scheme, the WC-NSGA-II and the PCSEA which are with the similar scheme are detailed further. For convenience of reviewing the related nadir point estimation methods, multiple fundamental concepts of ... |
Definition 1 Generally, there are [MATH] extreme points denoted as [MATH] [MATH] [MATH] in an [MATH] -objective optimization problem, |
[MATH] , and [MATH] where [MATH] and [MATH] Definition 2 The nadir point is defined as [MATH] , where [MATH] Definition 3 The worst point is defined as [MATH] , where [MATH] and [MATH] |
Definition 4 The ideal point is defined as [MATH] , where [MATH] and [MATH] Furthermore, the ideal point, extreme point, worst point, nadir point, and the PF are plotted with a bi-objective optimization problem in Fig. for intuitively understanding their significance. With these fundamental definitions, a couple of nad... |
WC-NSGA-II was designed based on NSGA-II by modifying its crowding distance assignment. According to the definition of nadir point in Definition , WC-NSGA-II naturally emphasized the solutions with maximal objectives front-wise. Specifically, solutions on a particular non-dominated front were sorted with an increasing ... |
PCSEA employed the corner-sorting to focus on the extreme points during the evolution. Specifically, there were [MATH] ascended lists during the executions of PCSEA. The first [MATH] lists were about the [MATH] objectives of the solutions, while the other [MATH] lists were about the excluded square [MATH] norm with eac... |
Briefly, major concerns in these two nadir point estimation algorithms are summarized as 1) over-emphasizing extreme points leads to the loss of diversity which in turn deteriorates the convergence of the found nadir points, and 2) simultaneously minimizing the objectives does not scale to problems with different objec... |
[EQUATION] where [MATH] is a factor with the value greater than [MATH] to highlight the priority of solving its associated term. In order to better justify our motivation, an example with bi-objective is plotted in Fig. in which [MATH] are the extreme points, [MATH] are the worse points, and the shaded region denotes t... |
II-B IGD -EMOA Prior to the introduction of IGD -EMOA, it is necessary to compare the differences between the IGD and IGD indicators. For this purpose, we first list their respective mathematical formulations. Then the superiority of IGD indicator is highlighted. Finally, the IGD -EMOA is discussed much further. |
Basically, the IGD indicator is with the form formulated by ( [EQUATION] where [MATH] denotes a set of reference points in the calculation of IGD, [MATH] denotes the non-dominated solutions [MATH] denotes the nearest distance from [MATH] to solutions in [MATH] , and the distance from [MATH] to the solution [MATH] in [M... |
that IGD cannot differentiate the quality of generated solutions when they are non-dominated to the solutions in [MATH] , and the IGD indicator is proposed by changing the calculation of [MATH] to [MATH] |
IGD -EMOA employed the IGD indicator as its selection mechanism. In addition, the [MATH] in IGD -EMOA is sampled from the approximate PF. Specifically, it supposed that the PF is obtained by solving [MATH] where [MATH] is from the non-dominated solutions of the current population. However, this approximate approach for... |
In summary, we first introduce details of the proposed DNPE which is motivated by the insightful observations in deficiencies of WC-NSGA-II and PCSEA. With the help of the estimated nadir point, the Utopian PF is constructed and the reference points are sampled for the calculation of the proposed MaOEA/IGD. Compared to... |
III Proposed algorithm In this section, the proposed Inverted Generational Distance indicator-based evolutionary algorithm for addressing many-objective optimization problems (in short for MaOEA/IGD) is presented. To be specific, the framework of the proposed algorithm is outlined first. Then the details of each step i... |
III-A Framework of the Proposed Algorithm Because the proposed algorithm is based on the IGD indicator, a set of uniformly distributed points which is [MATH] , which are evenly distributed in the Utopian PF is generated first (Subsection III-B ). Then the population with the predefined size is randomly initialized in t... |
[MATH] Uniformly generate reference points for IGD indicator; [MATH] Randomly initialize the population; Fitness evaluation on [MATH] |
[MATH] while stopping criteria are not satisfied do Assign the ranks and the proximity distances for individuals in [MATH] [MATH] Generate offspring from [MATH] |
Fitness evaluation on [MATH] Assign the rank and the proximity distance for each individual in [MATH] 10 [MATH] Environmental selection from [MATH] |
11 [MATH] 12 13 end while Return [MATH] Algorithm 1 Framework of the Proposed Algorithm III-B Uniformly Generating Reference Points |
In order to obtain the [MATH] , the extreme points of the problem [MATH] to be optimized are calculated first. Then the ideal point and the nadir point are extracted. Next, a set of solutions is uniformly sampled from the constrained [MATH] -dimensional hyperplane. Finally, these solutions are transformed into the Utop... |
Input: Optimization problem [MATH] ; the size [MATH] of [MATH] Output: [MATH] Estimate the extreme points of [MATH] with Algorithm |
[MATH] Extract the ideal point; [MATH] Extract the nadir point; [MATH] Uniformly generate [MATH] points from the constrained hyperplane; |
for [MATH] to [MATH] do for [MATH] to [MATH] do [MATH] [MATH] end for 10 end for Return [MATH] Algorithm 2 Uniformly Generate [MATH] for IGD Indicator |
To estimate the extreme points, the motivation mentioned in Subsection II-A is implemented, and the details are presented in Algorithm . Specifically, the [MATH] extreme points are estimated individually based on the [MATH] objectives of the optimization problem. Furthermore, to estimate the [MATH] -th extreme point [M... |
, which is widely used by some state-of-the-art MaOEAs, such as MOEA/D and NSGA-III , is employed for the generation of [MATH] (line of Algorithm ). Ultimately, all the points in [MATH] are transformed into the Utopian PF, which are detailed in lines of Algorithm |
Input: Optimization problem [MATH] Output: Extreme points [MATH] [MATH] for [MATH] to [MATH] do [MATH] for [MATH] to [MATH] do if |
[MATH] then [MATH] [MATH] end if end for 10 [MATH] 11 12 [MATH] 13 [MATH] [MATH] 14 end for Return [MATH] Algorithm 3 Estimate Extreme Points |
III-C Assigning Ranks and Proximity Distances When [MATH] has been generated, the population is randomly initialized in the feasible search space first, and then the fitness of individuals are evaluated. Next, the rank value and the proximity distances of each individual are assigned. It is noted here that, the rank va... |
Particularly, three rank values, denoted by [MATH] [MATH] , and [MATH] , exist in the proposed algorithm for all the individuals. Specifically, the way to rank individual [MATH] is based on the definitions given as follows. |
Definition 5 Individual [MATH] is ranked as [MATH] , if it dominates at least one solution in [MATH] Definition 6 Individual [MATH] is ranked as [MATH] , if it is non-dominated to all the solutions in [MATH] |
Definition 7 Individual [MATH] is ranked as [MATH] , if it is dominated by all the solutions in [MATH] , or dominated by a part of solutions in [MATH] but non-dominated to the remaining solutions. |
With Definitions , and , it is concluded that Pareto-optimal solutions are all with rank values [MATH] [MATH] , and [MATH] , if the PF is convex, a hyperplane, and concave, respectively . To be specific, if the PF of a minimization problem is a hyperplane, the Utopian PF is obviously equivalent to the PF. Consequently,... |
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