Natural Computing

Emmerich, Michael; Deutz, André; Yevseyeva, Iryna

doi: 10.1007/s11047-022-09928-9pmid: N/A

Lai, Leonardo; Fiaschi, Lorenzo; Cococcioni, Marco; Deb, Kalyanmoy

doi: 10.1007/s11047-022-09911-4pmid: N/A

This work aims at reviewing the state of the art of the field of lexicographic multi/many-objective optimization. The discussion starts with a review of the literature, emphasizing the numerous application in the real life and the recent burst received by the advent of new computational frameworks which work well in such contexts, e.g., Grossone Methodology. Then the focus shifts on a new class of problems proposed and studied for the first time only recently: the priority-levels mixed-pareto-lexicographic multi-objective-problems (PL-MPL-MOPs). This class of programs preserves the original preference ordering of pure many-objective lexicographic optimization, but instantiates it over multi-objective problems rather than scalar ones. Interestingly, PL-MPL-MOPs seem to be very well qualified for modeling real world tasks, such as the design of either secure or fast vehicles. The work also describes the implementation of an evolutionary algorithm able to solve PL-MPL-MOPs, and reports its performance when compared against other popular optimizers.

Irawan, Dani; Naujoks, Boris; Bäck, Thomas; Emmerich, Michael

doi: 10.1007/s11047-022-09910-5pmid: N/A

Optimization problems with multiple objectives and many input variables inherit challenges from both large-scale optimization and multi-objective optimization. To solve the problems, decomposition and transformation methods are frequently used. In this study, an improved control variable analysis is proposed based on dominance and diversity in Pareto optimization. Further, the decomposition method is used in a cooperative coevolution framework with orthogonal sampling mutation. The algorithm’s performances are compared against the weighted optimization framework. The results show that the proposed decomposition method has much better accuracy compared to the traditional method. The results also show that the cooperative coevolution framework with a good grouping is very competitive. Additionally, the number of search directions in orthogonal sampling can be easily configured. A small number of search directions will reduce the search space greatly while also restricting the area that can be explored and vice versa.

José-García, Adán; Handl, Julia

doi: 10.1007/s11047-022-09909-ypmid: N/A

The problem of cluster analysis eludes a unique mathematical definition. Instead, a variety of different instantiations of the problem can be defined using specific measures of internal cluster validity. In turn, such internal cluster validity measures rely on quantifying dissimilarity between entities. This article explores the interaction between dissimilarity measures and internal cluster validity techniques in the context of multi-objective clustering. It does so by contrasting two conceptually different approaches to multi-objective clustering: the multi-criterion clustering algorithm Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta$$\end{document}-MOCK, designed to optimise different measures of internal cluster validity over a single dissimilarity space, and the multi-view clustering algorithm MVMC, designed to optimise a single measure of internal cluster validity over distinct dissimilarity spaces. Our comparison highlights the interchangeable roles of distance functions and measures of internal cluster validity, which paves the way for the future design of a flexible, dual-purpose approach to multi-objective clustering.

Aspar, Pelin; Steinhoff, Vera; Schäpermeier, Lennart; Kerschke, Pascal; Trautmann, Heike; Grimme, Christian

doi: 10.1007/s11047-022-09919-wpmid: N/A

Single-objective continuous optimization can be challenging, especially when dealing with multimodal problems. This work sheds light on the effects that multi-objective optimization may have in the single-objective space. For this purpose, we examine the inner mechanisms of the recently developed sophisticated local search procedure SOMOGSA. This method solves multimodal single-objective continuous optimization problems based on first expanding the problem with an additional objective (e.g., a sphere function) to the bi-objective domain and subsequently exploiting local structures of the resulting landscapes. Our study particularly focuses on the sensitivity of this multiobjectivization approach w.r.t. (1) the parametrization of the artificial second objective, as well as (2) the position of the initial starting points in the search space. As SOMOGSA is a modular framework for encapsulating local search, we integrate Nelder–Mead local search as optimizer in the respective module and compare the performance of the resulting hybrid local search to its original single-objective counterpart. We show that the SOMOGSA framework can significantly boost local search by multiobjectivization. Hence, combined with more sophisticated local search and metaheuristics, this may help solve highly multimodal optimization problems in the future.

Zhu, Shuwei; Xu, Lihong; Goodman, Erik; Deb, Kalyanmoy; Lu, Zhichao

doi: 10.1007/s11047-022-09889-zpmid: N/A

In the last decade, it is widely known that the Pareto dominance-based evolutionary algorithms (EAs) are unable to deal with many-objective optimization problems (MaOPs) well, as it is hard to maintain a good balance between convergence and diversity. Instead, most researchers in this domain tend to develop EAs that do not rely on Pareto dominance (e.g., decomposition-based and indicator-based techniques) to solve MaOPs. However, it is still hard for these non-Pareto-dominance-based methods to solve MaOPs with unknown irregular PF shapes. In this paper, we develop a general framework for enhancing relaxed Pareto dominance methods to solve MaOPs, which can promote both convergence and diversity. During the environmental selection step, we use M different cases of relaxed Pareto dominance simultaneously, where each expands the dominance area of solutions for M-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M\,-$$\end{document} 1 objectives to improve the selection pressure, while the remaining one objective keeps unchanged. We conduct the experiments on a variety of test problems, the result shows that our proposed framework can obviously improve the performance of relaxed Pareto dominance in solving MaOPs, and is very competitive against or outperform some state-of-the-art many-objective EAs.

Weise, Jens; Mostaghim, Sanaz

doi: 10.1007/s11047-022-09908-zpmid: N/A

Pathfinding, also known as route planning, is one of the most important aspects of logistics, robotics, and other applications where engineers must balance many competing interests. There is a significant challenge in pathfinding problems with multiple objectives because many paths can map to the same objective value. Such multi-modal solutions cannot easily be found in multi-objective optimisation algorithms, which are typically geared towards selection mechanisms in the objective space. A niching approach for preserving good diverse solutions in the decision space is proposed in this paper, which is tailored for pathfinding problems. The criteria used to compare the solutions within the decision space are path similarity metrics, which we extend from a previous study, and are used instead of the well-established crowding distance. In two variations, we investigate the proposed meta-heuristic approach on a range of benchmark instances and compare the methodology to a deterministic optimisation approach.

Liu, Fei; Zhang, Qingfu; Han, Zhonghua

doi: 10.1007/s11047-022-09907-0pmid: N/A

In many real-world engineering design optimization problems, objective function evaluations are very time costly and often conducted by solving partial differential equations. Gradients of the objective functions can be obtained as a byproduct. Naturally, these problems can be solved more efficiently if gradient information is used. This paper studies how to do expensive multiobjective optimization when gradients are available. We propose a method, called MOEA/D–GEK, which combines MOEA/D and gradient-enhanced kriging. The gradients are used for building kriging models. Experimental studies on a set of test instances and an engineering problem of aerodynamic design optimization for a transonic airfoil show the high efficiency and effectiveness of our proposed method.

Javadi, Mahrokh; Mostaghim, Sanaz

doi: 10.1007/s11047-022-09921-2pmid: N/A

Many real-world multi-objective optimization problems inherently have multiple multi-modal solutions and it is in fact very important to capture as many of these solutions as possible. Several crowding distance methods have been developed in the past few years to approximate the optimal solution in the search space. In this paper, we discuss some of the shortcomings of the crowding distance-based methods such as inaccurate estimates of the density of neighboring solutions in the search space. We propose a new classification for the selection operations of Pareto-based multi-modal multi-objective optimization algorithms. This classification is based on utilizing nearby solutions from other fronts to calculate the crowding values. Moreover, to address some of the drawbacks of existing crowding methods, we propose two algorithms whose selection mechanisms are based on each of the introduced types of selection operations. These algorithms are called NxEMMO and ES-EMMO. Our proposed algorithms are evaluated on 14 test problems of various complexity levels. According to our results, in most cases, the NxEMMO algorithm with the proposed selection mechanism produces more diverse solutions in the search space in comparison to other competitive algorithms.

Mortveit, Henning S.

doi: 10.1007/s11047-023-09944-3pmid: N/A

Asynchronous, finite dynamical systems are constructed by composing vertex functions f1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_1$$\end{document}, f2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_2$$\end{document} through fn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_n$$\end{document} using an order π\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi$$\end{document} on {1,2,…,n}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{1,2,\ldots ,n\}$$\end{document} to form maps fπ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_\pi$$\end{document}. This article provides a systematic review and new results for comparing the dynamics of maps fπ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_\pi$$\end{document} and fπ′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_{\pi '}$$\end{document} formed using different composition orders π\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi$$\end{document} and π′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi '$$\end{document}. Comparisons are done at the level of (i) entire phase spaces (functional equivalence), (ii) the periodic orbits (cycle equivalence), and (iii) topological conjugation (dynamical equivalence). Conditions and criteria are given in terms of the dependency graph G of the vertex functions along with associated structures such as the acyclic orientations of G and the automorphism group of G. For each type of equivalence, graph-theoretic measures are provided that give an upper bound for the number of distinct structures that can be observed up to the respective notion of equivalence. Several connections to existing mathematical theory (e.g., combinatorics and graphs) are presented along with open questions.