An objective normalization strategy is essential in any evolutionary multiobjective or many-objective optimization (EMO or EMaO) algorithm, due to the distance calculations between objective vectors required to compute diversity and convergence of population members. For the decomposition-based EMO/EMaO algorithms involving the Penalty Boundary Intersection (PBI) metric, normalization is an important matter due to the computation of two distance metrics. In this article, we make a theoretical analysis of the effect of instabilities in the normalization process on the performance of PBI-based MOEA/D and a proposed PBI-based NSGA-III procedure. Although the effect is well recognized in the literature, few theoretical studies have been done so far to understand its true nature and the choice of a suitable penalty parameter value for an arbitrary problem. The developed theoretical results have been corroborated with extensive experimental results on three to 15-objective convex and non-convex instances of DTLZ and WFG problems. The article, makes important theoretical conclusions on PBI-based decomposition algorithms derived from the study.

Multiobjective evolutionary algorithms (MOEAs) have progressed significantly in recent decades, but most of them are designed to solve unconstrained multiobjective optimization problems. In fact, many real-world multiobjective problems contain a number of constraints. To promote research on constrained multiobjective optimization, we first propose a problem classification scheme with three primary types of difficulty, which reflect various types of challenges presented by real-world optimization problems, in order to characterize the constraint functions in constrained multiobjective optimization problems (CMOPs). These are feasibility-hardness, convergence-hardness, and diversity-hardness. We then develop a general toolkit to construct difficulty adjustable and scalable CMOPs (DAS-CMOPs, or DAS-CMaOPs when the number of objectives is greater than three) with three types of parameterized constraint functions developed to capture the three proposed types of difficulty. In fact, the combination of the three primary constraint functions with different parameters allows the construction of a large variety of CMOPs, with difficulty that can be defined by a triplet, with each of its parameters specifying the level of one of the types of primary difficulty. Furthermore, the number of objectives in this toolkit can be scaled beyond three. Based on this toolkit, we suggest nine difficulty adjustable and scalable CMOPs and nine CMaOPs, to be called DAS-CMOP1-9 and DAS-CMaOP1-9, respectively. To evaluate the proposed test problems, two popular CMOEAs—MOEA/D-CDP (MOEA/D with constraint dominance principle) and NSGA-II-CDP (NSGA-II with constraint dominance principle) and two popular constrained many-objective evolutionary algorithms (CMaOEAs)—C-MOEA/DD and C-NSGA-III—are used to compare performance on DAS-CMOP1-9 and DAS-CMaOP1-9 with a variety of difficulty triplets, respectively. The experimental results reveal that mechanisms in MOEA/D-CDP may be more effective in solving convergence-hard DAS-CMOPs, while mechanisms of NSGA-II-CDP may be more effective in solving DAS-CMOPs with simultaneous diversity-, feasibility-, and convergence-hardness. Mechanisms in C-NSGA-III may be more effective in solving feasibility-hard CMaOPs, while mechanisms of C-MOEA/DD may be more effective in solving CMaOPs with convergence-hardness. In addition, none of them can solve these problems efficiently, which stimulates us to continue to develop new CMOEAs and CMaOEAs to solve the suggested DAS-CMOPs and DAS-CMaOPs.

The Steiner tree problem (STP) aims to determine some Steiner nodes such that the minimum spanning tree over these Steiner nodes and a given set of special nodes has the minimum weight, which is NP-hard. STP includes several important cases. The Steiner tree problem in graphs (GSTP) is one of them. Many heuristics have been proposed for STP, and some of them have proved to be performance guarantee approximation algorithms for this problem. Since evolutionary algorithms (EAs) are general and popular randomized heuristics, it is significant to investigate the performance of EAs for STP. Several empirical investigations have shown that EAs are efficient for STP. However, up to now, there is no theoretical work on the performance of EAs for STP. In this article, we reveal that the (1+1) EA achieves 3/2-approximation ratio for STP in a special class of quasi-bipartite graphs in expected runtime , where , , and are, respectively, the number of Steiner nodes, the number of special nodes, and the largest weight among all edges in the input graph. We also show that the (1+1) EA is better than two other heuristics on two GSTP instances, and the (1+1) EA may be inefficient on a constructed GSTP instance.