A multiobjective optimization problem involves several conflicting objectives and has a set of Pareto optimal solutions. By evolving a population of solutions, multiobjective evolutionary algorithms (MOEAs) are able to approximate the Pareto optimal set in a single run. MOEAs have attracted a lot of research effort during the last 20 years, and they are still one of the hottest research areas in the field of evolutionary computation. This paper surveys the development of MOEAs primarily during the last eight years. It covers algorithmic frameworks such as decomposition-based MOEAs (MOEA/Ds), memetic MOEAs, coevolutionary MOEAs, selection and offspring reproduction operators, MOEAs with specific search methods, MOEAs for multimodal problems, constraint handling and MOEAs, computationally expensive multiobjective optimization problems (MOPs), dynamic MOPs, noisy MOPs, combinatorial and discrete MOPs, benchmark problems, performance indicators, and applications. In addition, some future research issues are also presented.

Multiobjective evolutionary algorithms: A survey of the state of the art

Memetic computation is a paradigm that uses the notion of meme(s) as units of information encoded in computational representations for the purpose of problem-solving. It covers a plethora of potentially rich meme-inspired computing methodologies, frameworks and operational algorithms including simple hybrids, adaptive hybrids and memetic automaton. In this paper, a comprehensive multi-facet survey of recent research in memetic computation is presented.

A Multi-Facet Survey on Memetic Computation

The Hausdorff distance dH is a widely used tool to measure the distance between different objects in several research fields. Possible reasons for this might be that it is a natural extension of the well-known and intuitive distance between points and/or the fact that dH defines in certain cases a metric in the mathematical sense. In evolutionary multiobjective optimization (EMO) the task is typically to compute the entire solution set-the so-called Pareto set-respectively its image, the Pareto front. Hence, dH should, at least at first sight, be a natural choice to measure the performance of the outcome set in particular since it is related to the terms spread and convergence as used in EMO literature. However, so far, dH does not find the general approval in the EMO community. The main reason for this is that dH penalizes single outliers of the candidate set which does not comply with the use of stochastic search algorithms such as evolutionary strategies. In this paper, we define a new performance indicator, Δp, which can be viewed as an “averaged Hausdorff distance” between the outcome set and the Pareto front and which is composed of (slight modifications of) the well-known indicators generational distance (GD) and inverted generational distance (IGD). We will discuss theoretical properties of Δp (as well as for GD and IGD) such as the metric properties and the compliance with state-of-theart multiobjective evolutionary algorithms (MOEAs), and will further on demonstrate by empirical results the potential of Δp as a new performance indicator for the evaluation of MOEAs.

Using the Averaged Hausdorff Distance as a Performance Measure in Evolutionary Multiobjective Optimization

In recent years, the research community has witnessed an explosion of literature dealing with the mimicking of behavioral patterns and social phenomena observed in nature towards efficiently solving complex computational tasks. This trend has been especially dramatic in what relates to optimization problems, mainly due to the unprecedented complexity of problem instances, arising from a diverse spectrum of domains such as transportation, logistics, energy, climate, social networks, health and industry 4.0, among many others. Notwithstanding this upsurge of activity, research in this vibrant topic should be steered towards certain areas that, despite their eventual value and impact on the field of bio-inspired computation, still remain insufficiently explored to date. The main purpose of this paper is to outline the state of the art and to identify open challenges concerning the most relevant areas within bio-inspired optimization. An analysis and discussion are also carried out over the general trajectory followed in recent years by the community working in this field, thereby highlighting the need for reaching a consensus and joining forces towards achieving valuable insights into the understanding of this family of optimization techniques.

Bio-inspired computation: Where we stand and what's next

Traditional multiobjective evolutionary algorithms (MOEAs) consider multiple objectives as a whole when solving multiobjective optimization problems (MOPs). However, this consideration may cause difficulty to assign fitness to individuals because different objectives often conflict with each other. In order to avoid this difficulty, this paper proposes a novel coevolutionary technique named multiple populations for multiple objectives (MPMO) when developing MOEAs. The novelty of MPMO is that it provides a simple and straightforward way to solve MOPs by letting each population correspond with only one objective. This way, the fitness assignment problem can be addressed because the individuals' fitness in each population can be assigned by the corresponding objective. MPMO is a general technique that each population can use existing optimization algorithms. In this paper, particle swarm optimization (PSO) is adopted for each population, and coevolutionary multiswarm PSO (CMPSO) is developed based on the MPMO technique. Furthermore, CMPSO is novel and effective by using an external shared archive for different populations to exchange search information and by using two novel designs to enhance the performance. One design is to modify the velocity update equation to use the search information found by different populations to approximate the whole Pareto front (PF) fast. The other design is to use an elitist learning strategy for the archive update to bring in diversity to avoid local PFs. CMPSO is comprehensively tested on different sets of benchmark problems with different characteristics and is compared with some state-of-the-art algorithms. The results show that CMPSO has superior performance in solving these different sets of MOPs.

Multiple Populations for Multiple Objectives: A Coevolutionary Technique for Solving Multiobjective Optimization Problems

In this paper, we propose and investigate a new local search strategy for multiobjective memetic algorithms. More precisely, we suggest a novel iterative search procedure, known as the Hill Climber with Sidestep (HCS), which is designed for the treatment of multiobjective optimization problems, and show further two possible ways to integrate the HCS into a given evolutionary strategy leading to new memetic (or hybrid) algorithms. The pecularity of the HCS is that it is intended to be capable both moving toward and along the (local) Pareto set depending on the distance of the current iterate toward this set. The local search procedure utilizes the geometry of the directional cones of such optimization problems and works with or without gradient information. Finally, we present some numerical results on some well-known benchmark problems, indicating the strength of the local search strategy as a standalone algorithm as well as its benefit when used within a MOEA. For the latter we use the state of the art algorithms Nondominated Sorting Genetic Algorithm-II and Strength Pareto Evolutionary Algorithm 2 as base MOEAs.

HCS: A New Local Search Strategy for Memetic Multiobjective Evolutionary Algorithms

In this paper we propose a novel iterative search procedure for multi-objective optimization problems. The iteration process -- though derivative free -- utilizes the geometry of the directional cones of such optimization problems, and is capable both of moving toward and along the (local) Pareto set depending on the distance of the current iterate toward this set. Next, we give one possible way of integrating this local search procedure into a given EMO algorithm resulting in a novel memetic strategy. Finally, we present some numerical results on some well-known benchmark problems indicating the strength of both the local search strategy as well as the new hybrid approach.

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A new memetic strategy for the numerical treatment of multi-objective optimization problems

Batteries have enabled the use of portable electronic devices, including smartphones, laptops, tablets, e-book readers, and global positioning system (GPS) devices. This article documents the results from two surveys of 1,200 respondents from the United States intended to understand users' satisfaction, concerns, charging habits, and expectations with respect to the batteries in their portable electronic devices. The results are useful for requirements planning and comparing different portable electronic devices using the metrics of usage habits and batteryrelated expectations.

Batteries in Portable Electronic Devices: A User's Perspective

Multi-Objective Evolutionary Algorithms (MOEA) have been succesfully applied to solve control problems. However, many improvements are still to be accomplished. In this paper a new approach is proposed: the Multi-Objective Pole Placement with Evolutionary Algorithms (MOPPEA). The design method is based upon using complex-valued chromosomes that contain information about closed-loop poles, which are then placed through an output feedback controller. Specific cross-over and mutation operators were implemented in simple but efficient ways. The performance is tested on a mixed multi-objective H2/H∞ control problem.

Gustavo Sanchez

Papers

HCS: A New Local Search Strategy for Memetic Multiobjective Evolutionary Algorithms

A new memetic strategy for the numerical treatment of multi-objective optimization problems

Batteries in Portable Electronic Devices: A User's Perspective

Multi-objective pole placement with evolutionary algorithms

Solving Multi-Objective Linear Control Design Problems Using Genetic Algorithms