In evolutionary multiobjective optimization, maintaining a good balance between convergence and diversity is particularly crucial to the performance of the evolutionary algorithms (EAs). In addition, it becomes increasingly important to incorporate user preferences because it will be less likely to achieve a representative subset of the Pareto-optimal solutions using a limited population size as the number of objectives increases. This paper proposes a reference vector-guided EA for many-objective optimization. The reference vectors can be used not only to decompose the original multiobjective optimization problem into a number of single-objective subproblems, but also to elucidate user preferences to target a preferred subset of the whole Pareto front (PF). In the proposed algorithm, a scalarization approach, termed angle-penalized distance, is adopted to balance convergence and diversity of the solutions in the high-dimensional objective space. An adaptation strategy is proposed to dynamically adjust the distribution of the reference vectors according to the scales of the objective functions. Our experimental results on a variety of benchmark test problems show that the proposed algorithm is highly competitive in comparison with five state-of-the-art EAs for many-objective optimization. In addition, we show that reference vectors are effective and cost-efficient for preference articulation, which is particularly desirable for many-objective optimization. Furthermore, a reference vector regeneration strategy is proposed for handling irregular PFs. Finally, the proposed algorithm is extended for solving constrained many-objective optimization problems.

A Reference Vector Guided Evolutionary Algorithm for Many-Objective Optimization

It remains a challenge to identify a satisfactory set of tradeoff solutions for many-objective optimization problems that have more than three objectives. Coevolving the solutions with preference is becoming increasingly popular due to the enhanced local search capability, which makes it suitable for solving many-objective optimization problems. The framework of preference-inspired co-evolutionary algorithms (PICEAs) is suitable for obtaining promising performance for such problems, and the PICEA with goal vectors (PICEA-g) has achieved good performance in many applications. In this paper, an improved PICEA-g is proposed to further resolve this long-standing problem. The local principal component analysis operator is used as a controller to further expand the ability of the PICEA-g algorithm and enhance the convergence of PICEA-g. The proposed algorithm was evaluated using several widely used benchmark test suites that had 3–15 objectives and made a systematic comparison with five state-of-the-art multi-objective evolutionary algorithms. The resulting substantial amount of experimental results revealed that the algorithm we proposed could have good performance on most of the test suites assessed in our research, and it performs very well compared with other many-objective optimization algorithms. In addition, a sensitivity test was carried out to explore the impact of a key parameter in the algorithm we proposed in this study.

Preference-Inspired Co-Evolutionary Algorithms With Local PCA Oriented Goal Vectors for Many-Objective Optimization

Combining modified inverted generational distance indicator with reference-vector-guided selection for many-objective optimization

Preference-inspired coevolutionary algorithm with sparse autoencoder for many-objective optimization

Stochastic fractal search algorithm, a relatively efficient algorithm has certain advantages in solving multi-objective optimization problems. Decomposition-based strategy is effective for solving multi-objective optimization, which guarantees the diversity of solutions. On the other hand, a challenge for decomposition-based strategy is to enhance the algorithm’s convergence while maintaining the algorithm’s distribution. This paper proposes a reference vector-guided stochastic fractal search algorithm for multi-objective optimization(RVSFS). In the proposed algorithm, the leader solution of each subspace is selected by reference vectors to assist the generation of new solutions and ensure the convergence of the algorithm. A new selection criterion length–angle distance (LAD) is proposed. Furthermore, the Gaussian walk and Levy flight are conducted in different searching stages to balance the exploration and exploitation ability. To speed up convergence and maintain the diversity of solutions, an adaptive dynamic proportion mechanism is designed when using the random fractals and Gaussian walks to generate offspring. Since information is shared and exchanged among different solutions in each iteration, the searching process is accelerated and the exploration capability can be enhanced contemporary. On top of this, a vector-guided enhanced selection operator is designed in the elite selection step to provide desirable distribution. Our experimental results on 45 benchmarks from UF and WFG test suits show prominent competitiveness compared with five state-of-the-art algorithms. RVSFS is very competitive on all 9 UF benchmarks in terms of the metric inverted generational distance and also performs well on 14 of 36 benchmarks from WFG in terms of the metric inverted generational distance and the metric hypervolume. 

Adaptive stochastic fractal search algorithm for multi-objective optimization

The excellent performance of evolutionary multi-objective algorithms based on S metric selection (SMS) has been identified by many researchers. However, huge computational effort of S metric calculation has limited the full application of those algorithms. This paper proposes a novel S metric selection evolutionary algorithm (SMS-M2M) based on the population decomposition strategy MOEA/D-M2M. In SMS-M2M, SMS is conducted in each subpopulation instead of the whole population, which can avoid the S metric calculation of the total population. The purpose of population decomposition is to directly reduce the huge computational effort of calculating S metric and thus to give a simple but effective method to improve the effectiveness of SMS based algorithm. SMS-M2M utilizes the same SMS with a popular SMS based evolutionary algorithm SMS-EMOA. Numerical studies of SMS-M2M and SMS-EMOA have shown that the M2M population decomposition can effectively reduce the computational effort of SMS, meanwhile the theoretic analysis identifies the efficiency and effectiveness of SMS-M2M.

A Novel Evolutionary Multi-objective Algorithm Based on S Metric Selection and M2M Population Decomposition

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A Novel Evolutionary Multi-objective Algorithm Based on S Metric Selection and M2M Population Decomposition