mathematics Article A Mating Selection Based on Modified Strengthened Dominance Relation for NSGA-III Saykat Dutta 1 , Sri Srinivasa Raju M 1 , Rammohan Mallipeddi 2, *, Kedar Nath Das 1 and Dong-Gyu Lee 2   Citation: Dutta, S.; M, S.S.R.; Mallipeddi, R.; Das, K.N.; Lee, D.-G. A Mating Selection Based on Modified Strengthened Dominance Relation for NSGA-III. Mathematics 2021, 9, 2837. https://doi.org/ 10.3390/math9222837 Academic Editors: Theodore E. Simos and Charampos Tsitouras Received: 9 October 2021 Accepted: 3 November 2021 Published: 10 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics, National Institute of Technology Silchar, Assam 788010, India; saykat_rs@math.nits.ac.in (S.D.); m._rs@math.nits.ac.in (S.S.R.M.); kedarnath@math.nits.ac.in (K.N.D.) 2 Department of Artificial Intelligence, School of Electronics Engineering, Kyungpook National University, Daegu 41566, Korea; dglee@knu.ac.kr * Correspondence: mallipeddi@knu.ac.kr Abstract: In multi/many-objective evolutionary algorithms (MOEAs), to alleviate the degraded convergence pressure of Pareto dominance with the increase in the number of objectives, numerous modified dominance relationships were proposed. Recently, the strengthened dominance relation (SDR) has been proposed, where the dominance area of a solution is determined by convergence degree and niche size ( θ). Later, in controlled SDR (CSDR), θ and an additional parameter (k) associated with the convergence degree are dynamically adjusted depending on the iteration count. Depending on the problem characteristics and the distribution of the current population, different situations require different values of k, rendering the linear reduction of k based on the generation count ineffective. This is because a particular value of k is expected to bias the dominance relationship towards a particular region on the Pareto front (PF). In addition, due to the same reason, using SDR or CSDR in the environmental selection cannot preserve the diversity of solutions required to cover the entire PF. Therefore, we propose an MOEA, referred to as NSGA-III*, where (1) a modified SDR (MSDR)-based mating selection with an adaptive ensemble of parameter k would prioritize parents from specific sections of the PF depending on k, and (2) the traditional weight vector and non-dominated sorting-based environmental selection of NSGA-III would protect the solutions corresponding to the entire PF. The performance of NSGA-III* is favourably compared with state-of- the-art MOEAs on DTLZ and WFG test suites with up to 10 objectives. Keywords: convergence; decomposition; diversity; dominance; ensemble 1. Introduction In literature [1], evolutionary algorithms (EAs) have demonstrated their ability to tackle a variety of optimization problems efficiently. Many real-world optimization prob- lems involve several conflicting objectives that must be optimized simultaneously. Without prior preference information, the existence of conflicting objectives inevitably results in the impossibility of finding a single solution that is globally optimal concerning all of the objectives. In such a situation, instead of total order between various solutions, only partial orders between different solutions may be anticipated, resulting in a solution set consisting of a suite of alternative solutions that have been differently compromised. However, one of the difficulties in multi-objective optimization, compared to the single objective optimiza- tion, is that there does not exist a unique or straightforward quality assessment method to classify all the solutions obtained and to guide the search process towards better regions. Multi-objective evolutionary algorithms (MOEAs) are particularly suitable for this task because they simultaneously evolve a population of potential solutions to the problem at hand, which facilitates the search for a set of Pareto non-dominated solutions in a single run of the algorithm. Classically, a multi-objective problem (MOP) can be briefly stated as min f ( x)=( f 1 ( x), f 2 ( x),..., f M ( x)) s.t. x =( x 1 , x 2 ,..., x n ) ∈ X, X ⊆ R n (1) Mathematics 2021, 9, 2837. https://doi.org/10.3390/math9222837 https://www.mdpi.com/journal/mathematics