Multi-Objective Evolutionary Algorithms

Optimization problems arise in various fields of science, engineering, and industry. In many occasions, such optimization problems, particularly in the present scenario, involve a variety of decision variables and complex structured objectives, and constraints. Often, the classical or traditional optimization techniques face difficulty in solving such real world optimization problems in their original form. Due to deficiencies of classical optimization algorithms in solving large-scale, highly non-linear, and often non-differentiable problems, there is a need to develop efficient and robust computational algorithms, which can solve problems, numerically irrespective of their sizes. Taking inspiration from nature to develop computationally efficient algorithms is one way to deal with real world optimization problems. Broadly, one can put these algorithms in the field of computational sciences and in particular, to computational intelligence. Formally, computational intelligence (CI) is a set of nature-inspired computational methodologies and approaches to solve complex real world problems. The major constituents of CI are Fuzzy Systems (FS), Neural Networks (NN), and Swarm Intelligence (SI) and Evolutionary Computation (EC). Computational intelligence techniques are powerful, efficient, flexible, and reliable. Swarm Intelligence and Evolutionary Computation are two very useful components of computational intelligence that are primarily used to solve optimization problems. This book primarily concerns with various swarm and evolutionary optimization algorithms. This chapter provides a brief introduction to swarm and evolutionary algorithms.

Conventional evolutionary algorithms operate in a fixed search space with limiting parameter range, which is often predefined via a priori knowledge or trial and error in order to `guess' a suitable region comprising the global optimal solution. This requirement is hard, if not impossible, to fulfil in many real-world optimization problems since there is often no clue of where the desired solutions are located in these problems. Thus, this paper proposes an inductive---deductive learning approach for single- and multi-objective evolutionary optimization. The method is capable of directing evolution towards more promising search regions even if these regions are outside the initial predefined space. For problems where the global optimum is included in the initial search space, it is capable of shrinking the search space dynamically for better resolution in genetic representation to facilitate the evolutionary search towards more accurate optimal solutions. Validation results based on benchmark optimization problems show that the proposed inductive---deductive learning is capable of handling different fitness landscapes as well as distributing nondominated solutions uniformly along the final trade-offs in multi-objective optimization, even if there exist many local optima in a high-dimensional search space or the global optimum is outside the predefined search region.

Evolutionary algorithms have been widely used to tackle multi-objective optimization problems. Despite many successful applications there is a lack of theoretical knowledge on how and why this type of algorithm works particularly well for multi-objective problems. This special issue presents current advances to the theoretical understanding of evolutionary algorithms for multi-objective optimization. It comprises four very high quality papers which have been selected via a rigorous reviewing process. The first paper “On the Effect of Populations in Evolutionary Multi-Objective Optimisation” by Oliver Giel and Per Kristian Lehre studies the basic question whether a population is provably useful to compute the whole Pareto front of a multi-objective optimization problem. They compare restart strategies of several single-objective approaches based on one individual to a natural multi-objective approach using a population. Their analyses point out the benefits of the multi-objective evolutionary algorithm in a rigorous way. Using illustrative example functions they provide not only proven results but also a good intuition about why population-based approaches can outperform individual-based approaches in multi-objective optimization. The paper “Exploring the Runtime of an Evolutionary Algorithm for the Multi-Objective Shortest Path Problem” by Christian Horoba examines the behavior of evolutionary algorithms for the classical NP-hard multi-objective shortest path problem. The author considers an evolutionary approach based on -dominance and proves that this algorithm constitutes a fully polynomial-time randomized approximation scheme (FPRAS) for the problem. The algorithm achieves this by mimicking a problemspecific algorithm for that problem. This provides general insight into why evolutionary algorithms can perform well for problems that allow for efficient problem-specific algorithms. Hypervolume-based algorithms have become very popular in evolutionary multiobjective optimization. In the paper, “An Efficient Algorithm for Computing Hypervolume Contributions,” Karl Bringmann and Tobias Friedrich present a faster algorithm for determining the individual with the smallest hypervolume contribution in a given population. Furthermore, they study the problem of removing λ > 1 individuals with the smallest hypervolume contribution from a given population and present an elegant and fast method to carry out this task.

Bilevel optimization problems require every feasible upper-level solution to satisfy optimality of a lower-level optimization problem. These problems commonly appear in many practical problem solving tasks including optimal control, process optimization, game-playing strategy development, transportation problems, and others. In the context of a bilevel single objective problem, there exists a number of theoretical, numerical, and evolutionary optimization results. However, there does not exist too many studies in the context of having multiple objectives in each level of a bilevel optimization problem. In this paper, we address bilevel multi-objective optimization issues and propose a viable algorithm based on evolutionary multi-objective optimization (EMO) principles. Proof-of-principle simulation results bring out the challenges in solving such problems and demonstrate the viability of the proposed EMO technique for solving such problems. This paper scratches the surface of EMO-based solution methodologies for bilevel multi-objective optimization problems and should motivate other EMO researchers to engage more into this important optimization task of practical importance.

In today’s world, the interest on multi-objective optimization through evolutionary algorithms (EAs) is growing day by day. However, most of the relative researches are confined within small scale with relatively fewer number of decision variables limited within 30, though real-world multi-objective optimization problems deal with, most of the times, with more than hundred decision variables. Also, optimization with fully separable decision variables along with non-separable decision variables leads to more optimal solutions than dealing with only any one of the both. In this paper, we have proposed an algorithm, which deals with medium- to large-scale multi-objective decision variables, compares the optimal solutions of separable and non-separable decision variables, and accepts the one having most optimized decision. Here we have adopted the test functions (large-scale multiobjective and many-objective optimization test problems for separable decision variables and Zitzler–Deb–Thiele test suit for non-separable decision variables) that are scalable with more than 100 decision variables and can range the results of both separable and non-separable decision variables.

An evolutionary algorithm based on fully connected weight networks for mixed-variable multi-objective optimization

For the past decade or so, evolutionary multi-objective optimization (EMO) methodologies have earned wide popularity for solving complex practical optimization problems, simply due to their ability to find a representative set of Pareto-optimal solutions for mostly two, three, and some extent to four and five-objective optimization problems. Recently, emphasis has been made in addressing the decision-making activities in arriving at a single preferred solution. The multiple criteria decision making (MCDM) literature offers a number of possibilities for such a task involving user preferences which can be supplied in different forms. This paper presents an interactive methodology for finding a preferred set of solutions, instead of the complete Pareto-optimal frontier, by incorporating preference information of the decision maker. Particularly, we borrow the concept of light beam search and combine it with the NSGA-II procedure. The working of this procedure has been demonstrated on a set of test problems and on engineering design problems having two to ten objectives, where the obtained solutions are found to match with the true Pareto-optimal solutions. The results highlight the utility of this approach towards eventually facilitating a better and more reliable optimization-cum-decision-making task.

Many real-world optimization problems are characterized by multiple conflicting objectives, which are known as multi-objective optimization problems (MOPs). In the last two decades, evolutionary algorithms have shown promising performance in solving various MOPs, and a large number of multi-objective evolutionary algorithms (MOEAs) have been proposed. In order to determine the most suitable MOEA for a specific MOP, it is usually necessary to perform experiments to compare the performance of multiple candidate MOEAs. In 2017, an evolutionary multi-objective optimization platform was proposed by us, called PlatEMO, which provides the source codes of many state-of-the-art MOEAs and helps researchers perform batch experiments on these MOEAs. In this work, we illustrate the method of using the newest version of PlatEMO to solve MOPs in applications, by means of a case study on the feature selection problem, which is an important and difficult task in machine learning and data mining. This paper details the method of adding the feature selection problem to PlatEMO, and presents the experimental results of eight MOEAs on nine datasets in feature selection.

Aiming at the problems such as low computational efficiency and long process in dynamic topology optimization, combined with bi-directional evolutional structure optimization, the dynamic optimization problem is transformed into the static optimization problem in multiple load cases. Newmark method is used to analyze the structure dynamics load. The topology optimization model based on equivalent static load is established. On the basis of the equivalent static load, multi-objective dynamics evolutionary structure optimization is studied. And the mathematical model of the dynamic stiffness and natural frequency about the multi-objective optimization is established. The mathematical formula of the sensitivity is deduced. Then a complete optimization iterative process is given. The numerical example illustrates that it is feasible to solve the multi-objective dynamic topology optimization with equivalent static loads. Modal tracking technique is introduced to the multi-objective dynamic topology optimization. The numerical example shows that modal tracking technology can track the required modal in the optimization process.

Balancing exploration and exploitation with adaptive variation for evolutionary multi-objective optimization

Multi-objective optimization is widely found in many fields, such as logistics, economics, engineering, or whenever optimal decisions need to be made in the presence of tradeoff between two or more conflicting objectives. The synergy of probabilistic graphical approaches in evolutionary mechanism may enhance the iterative search process when interrelationships of the archived data has been learned, modeled, and used in the reproduction. This paper presents the implementation of probabilistic graphical approaches in solving multi-objective optimization problems under the evolutionary paradigm. First, the existing work on the synergy between probabilistic graphical models and evolutionary algorithms in the multi-objective framework will be presented. We will then show that the optimization problems can be solved using a restricted Boltzmann machine (RBM). The learning, modeling as well as sampling mechanisms of the RBM will be highlighted. Lastly, five studies that implement the RBM for solving multi-objective optimization problems will be discussed.

Multi-objective evolutionary optimization is widely utilized in industrial design. Despite the success of multi-objective evolutionary optimization algorithms in addressing complex optimization problems, research focusing on input disturbances remains limited. In many manufacturing processes, design parameters are vulnerable to random input disturbances, resulting in products that often perform less effectively than anticipated. To address this issue, we propose a novel robust multi-objective evolutionary optimization algorithm based on the concept of survival rate. The algorithm comprises two stages: the evolutionary optimization stage and the construction stage of the robust optimal front. In the former stage, we introduce the survival rate as a new optimization objective. Subsequently, we seek a robust optimal front that concurrently addresses convergence and robustness by employing a non-dominated sorting approach. Furthermore, we propose a precise sampling method and a random grouping mechanism to accurately recover solutions resilient to real noise while ensuring population’s diversity. In the latter stage, we introduce a performance measure that integrates both robustness and convergence to guide the construction of the robust optimal front. Experimental results demonstrate the superiority of the proposed algorithm in terms of both convergence and robustness compared to existing approaches under noisy conditions.

Proper planning of charging infrastructure can significantly facilitate the popularization of electric vehicles and alleviate users’ mileage anxiety. Charging station siting and sizing are two key challenges in the planning with each of them being a complex optimization problem. In this paper, a multi-objective optimization approach is proposed to solve them together. First, considering that accurate charging demand estimation is crucial for planning, a traffic road network is established for this purpose. A Monte Carlo method is used to estimate the spatiotemporal distribution of charging demand in a region based on the probabilistic characteristics of user trips. Since uncoordinated charging not only increases the load but also leads to unstable operation of the local power system, a heuristic algorithm is proposed to coordinate charging scheduling. Then, based on the scheduled demand, this paper proposes a framework for the siting and sizing of charging stations to optimize the benefits for both operators and users by minimizing the construction, operation and maintenance costs, and the user’s detour time. As the given problem is a complex multi-objective combinatorial optimization problem, it is easy to fall into local optimum if traditional evolutionary algorithms are employed. Therefore, a multi-objective dynamic binary particle swarm optimization method is designed to solve this problem effectively. Finally, experimental simulations show that the proposed method outperforms the other comparative algorithms in terms of solution quality. A case study is presented to demonstrate the applicability and effectiveness of the proposed method in optimizing the location and capacity of charging stations.

Some real-world optimization problems involve multiple decision makers holding different positions, each of whom has multiple conflicting objectives. These problems are defined as multiparty multiobjective optimization problems (MPMOPs). Although evolutionary multiobjective optimization has been widely studied for many years, little attention has been paid to multiparty multiobjective optimization in the field of evolutionary computation. In this paper, a class of MPMOPs, that is, MPMOPs having common Pareto optimal solutions, is addressed. A benchmark for MPMOPs, obtained by modifying an existing dynamic multiobjective optimization benchmark, is provided, and a multiparty multiobjective evolutionary algorithm to find the common Pareto optimal set is proposed. The results of experiments conducted using the benchmark show that the proposed multiparty multiobjective evolutionary algorithm is effective.

A surrogate-assisted multi-objective evolutionary algorithm with dimension-reduction for production optimization