A Cooperative Co-evolutionary Approach for Multi-objective Optimization

In large-scale multiobjective optimization, too many decision variables hinder the convergence search of evolutionary algorithms. Reducing the search range of the decision space will significantly alleviate this puzzle. With this in mind, this paper proposes a fuzzy decision variables framework for large-scale multiobjective optimization. The framework divides the entire evolutionary process into two main stages: fuzzy evolution and precise evolution. In fuzzy evolution, we blur the decision variables of the original solution to reduce the search range of the evolutionary algorithm in the decision space so that the evolutionary population can quickly converge. The degree of fuzzification gradually decreases with the evolutionary process. Once the population approximately converges, the framework will turn to precise evolution. In precise evolution, the actual decision variables of the solution are directly optimized to increase the diversity of the population so as to be closer to the true Pareto optimal front. Finally, this paper embeds some representative algorithms into the proposed framework and verifies the framework’s effectiveness through comparative experiments on various large-scale multiobjective problems with 500 to 5000 decision variables. Experimental results show that in large-scale multiobjective optimization, the framework proposed in this paper can significantly improve the performance and computational efficiency of multiobjective optimization algorithms.

A fast sampling based evolutionary algorithm for million-dimensional multiobjective optimization

For large-scale multi-objective optimization, it is particularly challenging for evolutionary algorithms to converge to the Pareto Front. Most existing multi-objective evolutionary algorithms (MOEAs) handle convergence and diversity in a mutually dependent manner during the evolution process. In this case, the performance degradation of one solution may lead to the deterioration of the performance of the other solution. This paper proposes a two-stage multi-objective optimization algorithm based on decision variable clustering (LSMOEA-VT) to solve large-scale optimization problems. In LSMOEA-VT, decision variables are divided into two categories and use dimensionality reduction methods to optimize the variables that affect evolutionary convergence. Following this, we performed an interdependence analysis to break down the convergence variables into multiple subcomponents that are more tractable. Furthermore, a non-dominated dynamic weight aggregation method is used to enhance the diversity of the population. To evaluate the performance of our proposed algorithm, we performed extensive comparative experiments against four optimization algorithms across a diverse set of benchmark problems, including eight multi-objective optimization problems and nine large-scale optimization problems. The experimental results show that our proposed algorithm performs well on some test functions and has certain competitiveness.

In multiobjective optimization, it is nontrivial for decision makers to articulate preferences without a priori knowledge, which is particularly true when the number of objectives becomes large. Depending on the shape of the Pareto front, optimal solutions such as knee points may be of interest. Although several multi- and many-objective optimization test suites have been proposed, little work has been reported focusing on designing multiobjective problems whose Pareto front contains complex knee regions. Likewise, few performance indicators dedicated to evaluate an algorithm's ability of accurately locating all knee points in high-dimensional objective space have been suggested. This paper proposes a set of multiobjective optimization test problems whose Pareto front consists of complex knee regions, aiming to assess the capability of evolutionary algorithms to accurately identify all knee points. Various features related to knee points have been taken into account in designing the test problems, including symmetry, differentiability, and degeneration. These features are also combined with other challenges in solving the optimization problems, such as multimodality, linkage between decision variables, nonuniformity, and scalability of the Pareto front. The proposed test problems are scalable to both decision and objective spaces. Accordingly, new performance indicators are suggested for evaluating the capability of optimization algorithms in locating the knee points. The proposed test problems, together with the performance indicators, offer a new means to develop and assess preference-based evolutionary algorithms for solving multi- and many-objective optimization problems.

A multi-objective particle swarm optimization (MOPSO) approach for multi-objective economic load dispatch problem in power system is presented in this paper. The economic load dispatch problem is a non-linear constrained multi-objective optimization problem. The proposed MOPSO approach handles the problem as a multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives and has a diversity-preserving mechanism using an external memory (call “repository”) and a geographically-based approach to find widely different Pareto-optimal solutions. In addition, fuzzy set theory is employed to extract the best compromise solution. Several optimization runs of the proposed MOPSO approach were carried out on the standard IEEE 30-bus test system. The results revealed the capabilities of the proposed MOPSO approach to generate well-distributed Pareto-optimal non-dominated solutions of multi-objective economic load dispatch. Comparison with Multi-objective Evolutionary Algorithm (MOEA) showed the superiority of the proposed MOPSO approach and confirmed its potential for solving multi-objective economic load dispatch.

Recently, a number of studies have attempted to characterize the interaction between objectives and decision variables in multiobjective problems. In this paper, we continue this line of research by focusing specifically on quantifying observable differences in the ease of optimizing extreme solutions (i.e. solutions on the limits of the Pareto front). We propose an evolutionary path length correlation (EPLC) measurement in the decision variable space that is computed by tracing the evolutionary history of solutions on the Pareto front. We draw on the length scale measure and extend the well-known fitness distance correlation to multiobjective optimization problems. Here, the overarching goal is to investigate the emergent dynamics of specific problem–algorithm combinations, rather than the characterization of the search landscape per se. We evaluate the efficacy of the EPLC using benchmark continuous multiobjective problems and combinatorial problems with controllable objective interactions and known Pareto optima. In some problems, observable differences in the convergence to extreme solutions in each objective can be captured using the EPLC. Our results go some way towards furthering our understanding of how specific algorithms traverse the landscape, given interactions between both decision variables and objectives.KeywordsMultiobjective optimizationFitness distance correlationLength scale

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.

Naphtha pyrolysis is one of the important routes for simultaneous production of ethylene and propylene. With recent increase in demand of both ethylene and propylene, understanding of naphtha pyrolysis becomes important for producing with increasing yield of these valuable products. Simultaneous maximization of yield of these two products is mathematically formulated as a multiobjective optimization (MOO) problem. Improved selected scheme is incorporated in the existing multiobjective differential evolution (MODE) algorithm and a new evolutionary algorithm is proposed. Both the evolutionary algorithms [i.e., MODE III and MODE-III with improved selection scheme (MODE III-ISS)] are used for MOO of industrial naphtha cracker unit. Two objectives (maximization of ethylene yield and propylene yield) and decision variables [pressure of the reactor tube (P), temperature of the reactor (T), initial flow rate of naphtha (F 0), and steam to naphtha ratio (SOR)] are considered for MOO study. MODE III and MODE III-ISS algorithms results are compared and presented, which clearly shows that the proposed MODE III-ISS algorithm possesses certain advantages over the MODE III algorithm (such as number of successful selections and percentage convergence with respect to initial number of population points, the quality of the obtained nondominated [ND] solutions).

Real world optimization problems are often too complex to be solved through analytical means. Evolutionary algorithms, a class of algorithms that borrow paradigms from nature, are particularly well suited to address such problems. These algorithms are stochastic methods of optimization that have become immensely popular recently, because they are derivative-free methods, are not as prone to getting trapped in local minima (as they are population based), and are shown to work well for many complex optimization problems. Although evolutionary algorithms have conventionally focussed on optimizing single objective functions, most practical problems in engineering are inherently multi-objective in nature. Multi-objective evolutionary optimization is a relatively new, and rapidly expanding area of research in evolutionary computation that looks at ways to address these problems. In this chapter, we provide an overview of some of the most significant issues in multi-objective optimization (Deb, 2001).

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

In this paper, we propose a framework to accelerate the computational efficiency of evolutionary algorithms on large-scale multiobjective optimization. The main idea is to track the Pareto optimal set (PS) directly via problem reformulation. To begin with, the algorithm obtains a set of reference directions in the decision space and associates them with a set of weight variables for locating the PS. Afterwards, the original large-scale multiobjective optimization problem is reformulated into a low-dimensional single-objective optimization problem. In the reformulated problem, the decision space is reconstructed by the weight variables and the objective space is reduced by an indicator function. Thanks to the low dimensionality of the weight variables and reduced objective space, a set of quasi-optimal solutions can be obtained efficiently. Finally, a multiobjective evolutionary algorithm is used to spread the quasi-optimal solutions over the approximate Pareto optimal front evenly. Experiments have been conducted on a variety of large-scale multiobjective problems with up to 5000 decision variables. Four different types of representative algorithms are embedded into the proposed framework and compared with their original versions, respectively. Furthermore, the proposed framework has been compared with two state-of-the-art algorithms for large-scale multiobjective optimization. The experimental results have demonstrated the significant improvement benefited from the framework in terms of its performance and computational efficiency in large-scale multiobjective optimization.

Constrained multiobjective optimization problems (CMOPs) widely exist in real-world applications, and they are challenging for conventional evolutionary algorithms (EAs) due to the existence of multiple constraints and objectives. When the number of objectives or decision variables is scaled up in CMOPs, the performance of EAs may degenerate dramatically and may fail to obtain any feasible solutions. To address this issue, we propose a paired offspring generation-based multiobjective EA for constrained large-scale optimization. The general idea is to emphasize the role of offspring generation in reproducing some promising feasible or useful infeasible offspring solutions. We first adopt a small set of reference vectors for constructing several subpopulations with a fixed number of neighborhood solutions. Then, a pairing strategy is adopted to determine some pairwise parent solutions for offspring generation. Consequently, the pairwise parent solutions, which could be infeasible, may guide the generation of well-converged solutions to cross the infeasible region(s) effectively. The proposed algorithm is evaluated on CMOPs with up to 1000 decision variables and ten objectives. Moreover, each component in the proposed algorithm is examined in terms of its effect on the overall algorithmic performance. Experimental results on a variety of existing and our tailored test problems demonstrate the effectiveness of the proposed algorithm in constrained large-scale multiobjective optimization.

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

Most evolutionary multiobjective optimization algorithms are designed to approximate the entire Pareto front. During the last decade, a series of preference-based evolutionary algorithms have been developed, where a part of the Pareto front is approximated by incorporating the preferences of a Decision Maker. However, only a few such algorithms are able to obtain well-distributed solutions covering the complete “region of interest” that is determined by a reference point. In this paper, a preference-based evolutionary algorithm for approximating the region of interest is proposed. It is based on the state-of-the-art genetic algorithm NSGA-II and the CHIM approach introduced in the NBI method which is used to obtain uniformly distributed solutions in the region of interest. The efficiency of the proposed algorithm has been experimentally evaluated and compared to other state-of-the-art multiobjective preference-based evolutionary algorithms by solving a set of multiobjective optimization benchmark problems. It has been shown that the incorporation of the Decision Maker’s preferences and the CHIM approach into the NSGA-II algorithm allows approximating the whole region of interest accurately while maintaining a good distribution of the obtained solutions.

There can be a complicated mapping relation between decision variables and objective functions in multi-objective optimization problems (MOPs). It is uncommon that decision variables influence objective functions equally. Decision variables act differently in different objective functions. Hence, often, the mapping relation is unbalanced, which causes some redundancy during the search in a decision space. In response to this scenario, we propose a novel memetic (multi-objective) optimization strategy based on dimension reduction in decision space (DRMOS). DRMOS firstly analyzes the mapping relation between decision variables and objective functions. Then, it reduces the dimension of the search space by dividing the decision space into several subspaces according to the obtained relation. Finally, it improves the population by the memetic local search strategies in these decision subspaces separately. Further, DRMOS has good portability to other multi-objective evolutionary algorithms (MOEAs); that is, it is easily compatible with existing MOEAs. In order to evaluate its performance, we embed DRMOS in several state of the art MOEAs to facilitate our experiments. The results show that DRMOS has the advantage in terms of convergence speed, diversity maintenance, and portability when solving MOPs with an unbalanced mapping relation between decision variables and objective functions.