A federated data-driven evolutionary algorithm for expensive multi-/many-objective optimization

Data-driven optimization has found many successful applications in the real world and received increased attention in the field of evolutionary optimization. Most existing algorithms assume that the data used for optimization are always available on a central server for construction of surrogates. This assumption, however, may fail to hold when the data must be collected in a distributed way and are subject to privacy restrictions. This paper aims to propose a federated data-driven evolutionary multi-/many-objective optimization algorithm. To this end, we leverage federated learning for surrogate construction so that multiple clients collaboratively train a radial-basis-function-network as the global surrogate. Then a new federated acquisition function is proposed for the central server to approximate the objective values using the global surrogate and estimate the uncertainty level of the approximated objective values based on the local models. The performance of the proposed algorithm is verified on a series of multi-/many-objective benchmark problems by comparing it with two state-of-the-art surrogate-assisted multi-objective evolutionary algorithms.

Multi-objective particle swarm optimization with R2 indicator and adaptive method

Multi-objective particle swarm optimization algorithms encounter significant challenges when tackling many-objective optimization problems. This is mainly because of the imbalance between convergence and diversity that occurs when increasing the selection pressure. In this paper, a novel adaptive MOPSO (ANMPSO) algorithm based on R2 contribution and adaptive method is developed to improve the performance of MOPSO. First, a new global best solutions selection mechanism with R2 contribution is introduced to select leaders with better diversity and convergence. Second, to obtain a uniform distribution of particles, an adaptive method is used to guide the flight of particles. Third, a re-initialization strategy is proposed to prevent particles from trapping into local optima. Empirical studies on a large number (64 in total) of problem instances have demonstrated that ANMPSO performs well in terms of inverted generational distance and hyper-volume metrics. Experimental studies on the practical application have also revealed that ANMPSO could effectively solve problems in the real world.

A Polar-Metric-Based Evolutionary Algorithm

Over the past two decades, numerous multi- and many-objective evolutionary algorithms (MOEAs and MaOEAs) have been proposed to solve the multi- and many-objective optimization problems (MOPs and MaOPs), respectively. It is known that the difficulty of maintaining the convergence and diversity performances rapidly grows as the number of objectives increases. This phenomenon is especially evident for the Pareto-dominance-based EAs, because the nondominated sorting often fails to provide enough convergent pressure toward the Pareto front (PF). Therefore, many researchers came up with some non-Pareto-dominance-based EAs, which are based on indicator, decomposition, and so on. In this article, we propose a polar-metric ( p -metric)-based EA (PMEA) for tackling both MOPs and MaOPs. p -metric is a recently proposed performance indicator which adopts a set of uniformly distributed direction vectors. In PMEA, we use a two-phase selection which combines both nondominated sorting and p -metric. Moreover, a modification is proposed to adjust the direction vectors of p -metric dynamically. In the experiments, PMEA is compared with six state-of-the-art EAs in total and is measured by three performance metrics, including p -metric. According to the empirical results, PMEA shows promising performances on most of the test problems, involving both MOPs and MaOPs.

Knowledge-guided multiobjective particle swarm optimization with fusion learning strategies

Multiobjective particle swarm optimization (MOPSO) algorithm faces the difficulty of prematurity and insufficient diversity due to the selection of inappropriate leaders and inefficient evolution strategies. Therefore, to circumvent the rapid loss of population diversity and premature convergence in MOPSO, this paper proposes a knowledge-guided multiobjective particle swarm optimization using fusion learning strategies (KGMOPSO), in which an improved leadership selection strategy based on knowledge utilization is presented to select the appropriate global leader for improving the convergence ability of the algorithm. Furthermore, the similarity between different individuals is dynamically measured to detect the diversity of the current population, and a diversity-enhanced learning strategy is proposed to prevent the rapid loss of population diversity. Additionally, a maximum and minimum crowding distance strategy is employed to obtain excellent nondominated solutions. The proposed KGMOPSO algorithm is evaluated by comparisons with the existing state-of-the-art multiobjective optimization algorithms on the ZDT and DTLZ test instances. Experimental results illustrate that KGMOPSO is superior to other multiobjective algorithms with regard to solution quality and diversity maintenance.

Manifold Learning-Inspired Mating Restriction for Evolutionary Multiobjective Optimization With Complicated Pareto Sets

Under certain smoothness assumptions, the Pareto set of a continuous multiobjective optimization problem is a piecewise continuous manifold in the decision space, which can be derived from the Karush-Kuhn-Tucker condition. Despite that a number of multiobjective evolutionary algorithms (MOEAs) have been proposed, their performance on multiobjective optimization problems with complicated Pareto sets (MOP-cPS) is still unsatisfying. In this article, we adopt the concept of manifold and propose a manifold learning-inspired mating strategy to enhance the diversity maintenance in MOEAs for solving MOP-cPS efficiently. In the proposed strategy, all of the individuals are first clustered into different manifolds according to their distribution in the objective space, and then the mating reproduction is restricted among individuals in the same manifold. Moreover, we embed the proposed mating strategy in three representative MOEAs and compare the embedded MOEAs with their original versions using the assortative genetic operators on a variety of MOP-cPS. The experimental results demonstrate the significant performance improvements benefitting from the proposed mating restriction strategy.

Deep Reinforcement Learning for Multiobjective Optimization

This article proposes an end-to-end framework for solving multiobjective optimization problems (MOPs) using deep reinforcement learning (DRL), that we call DRL-based multiobjective optimization algorithm (DRL-MOA). The idea of decomposition is adopted to decompose the MOP into a set of scalar optimization subproblems. Then, each subproblem is modeled as a neural network. Model parameters of all the subproblems are optimized collaboratively according to a neighborhood-based parameter-transfer strategy and the DRL training algorithm. Pareto-optimal solutions can be directly obtained through the trained neural-network models. Specifically, the multiobjective traveling salesman problem (MOTSP) is solved in this article using the DRL-MOA method by modeling the subproblem as a Pointer Network. Extensive experiments have been conducted to study the DRL-MOA and various benchmark methods are compared with it. It is found that once the trained model is available, it can scale to newly encountered problems with no need for retraining the model. The solutions can be directly obtained by a simple forward calculation of the neural network; thereby, no iteration is required and the MOP can be always solved in a reasonable time. The proposed method provides a new way of solving the MOP by means of DRL. It has shown a set of new characteristics, for example, strong generalization ability and fast solving speed in comparison with the existing methods for multiobjective optimizations. The experimental results show the effectiveness and competitiveness of the proposed method in terms of model performance and running time.

Solving Large-Scale Multiobjective Optimization Problems With Sparse Optimal Solutions via Unsupervised Neural Networks

Due to the curse of dimensionality of search space, it is extremely difficult for evolutionary algorithms to approximate the optimal solutions of large-scale multiobjective optimization problems (LMOPs) by using a limited budget of evaluations. If the Pareto-optimal subspace is approximated during the evolutionary process, the search space can be reduced and the difficulty encountered by evolutionary algorithms can be highly alleviated. Following the above idea, this article proposes an evolutionary algorithm to solve sparse LMOPs by learning the Pareto-optimal subspace. The proposed algorithm uses two unsupervised neural networks, a restricted Boltzmann machine, and a denoising autoencoder to learn a sparse distribution and a compact representation of the decision variables, where the combination of the learnt sparse distribution and compact representation is regarded as an approximation of the Pareto-optimal subspace. The genetic operators are conducted in the learnt subspace, and the resultant offspring solutions then can be mapped back to the original search space by the two neural networks. According to the experimental results on eight benchmark problems and eight real-world problems, the proposed algorithm can effectively solve sparse LMOPs with 10000 decision variables by only 100000 evaluations.

Evolutionary Multiobjective Optimization Driven by Generative Adversarial Networks (GANs)

Recently, increasing works have been proposed to drive evolutionary algorithms using machine-learning models. Usually, the performance of such model-based evolutionary algorithms is highly dependent on the training qualities of the adopted models. Since it usually requires a certain amount of data (i.e., the candidate solutions generated by the algorithms) for model training, the performance deteriorates rapidly with the increase of the problem scales due to the curse of dimensionality. To address this issue, we propose a multiobjective evolutionary algorithm driven by the generative adversarial networks (GANs). At each generation of the proposed algorithm, the parent solutions are first classified into real and fake samples to train the GANs; then the offspring solutions are sampled by the trained GANs. Thanks to the powerful generative ability of the GANs, our proposed algorithm is capable of generating promising offspring solutions in high-dimensional decision space with limited training data. The proposed algorithm is tested on ten benchmark problems with up to 200 decision variables. The experimental results on these test problems demonstrate the effectiveness of the proposed algorithm.

Comparison of Transboundary Water Resources Allocation Models Based on Game Theory and Multi-Objective Optimization

Transboundary water resources allocation is an effective measure to resolve water-related conflicts. Aiming at the problem of water conflicts, we constructed water resources allocation models based on game theory and multi-objective optimization, and revealed the differences between the two models. We compare the Pareto front solved by the AR-MOEA method and the NSGA-II method, and analyzed the difference between the Nash–Harsanyi Leader–Follower game model and the multi-objective optimization model. The Huaihe River basin was selected as a case study. The results show that: (1) The AR-MOEA method is better than the NSGA-II method in terms of the diversity metric (Δ); (2) the solution of the asymmetric Nash–Harsanyi Leader–Follower game model is a non-dominated solution, and the asymmetric game model can obtain the same water resources allocation scheme of the multi-objective optimal allocation model under a specific preference structure; (3) after the multi-objective optimization model obtains the Pareto front, it still needs to construct the preference information of the Pareto front for a second time to make the optimal solution of a multi-objective decision, while the game model can directly obtain the water resources allocation scheme at one time by participating in the negotiation. The results expand the solution method of water resources allocation models and provide support for rational water resources allocation.

fSDE: efficient evolutionary optimisation for many-objective aero-engine calibration

Engine calibration aims at simultaneously adjusting a set of parameters to ensure the performance of an engine under various working conditions using an engine simulator. Due to the large number of engine parameters to be calibrated, the performance measurements to be considered, and the working conditions to be tested, the calibration process is very time-consuming and relies on the human knowledge. In this paper, we consider non-convex constrained search space and model a real aero-engine calibration problem as a many-objective optimisation problem. A fast many-objective evolutionary optimisation algorithm with shift-based density estimation, called fSDE, is designed to search for parameters with an acceptable performance accuracy and improve the calibration efficiency. Our approach is compared to several state-of-the-art many- and multi-objective optimisation algorithms on the well-known many-objective optimisation benchmark test suite and a real aero-engine calibration problem, and achieves superior performance. To further validate our approach, the studied aero-engine calibration is also modelled as a single-objective optimisation problem and optimised by some classic and state-of-the-art evolutionary algorithms, compared to which fSDE not only provides more diverse solutions but also finds solutions of high-quality faster.

On the Norm of Dominant Difference for Many-Objective Particle Swarm Optimization

Recent studies in multiobjective particle swarm optimization (PSO) have the tendency to employ Pareto-based technique, which has a certain effect. However, they will encounter difficulties in their scalability upon many-objective optimization problems (MaOPs) due to the poor discriminability of Pareto optimality, which will affect the selection of leaders, thereby deteriorating the effectiveness of the algorithm. This paper presents a new scheme of discriminating the solutions in objective space. Based on the properties of Pareto optimality, we propose the dominant difference of a solution, which can demonstrate its dominance in every dimension. By investigating the norm of dominant difference among the entire population, the discriminability between the candidates that are difficult to obtain in the objective space is obtained indirectly. By integrating it into PSO, we gained a novel algorithm named many-objective PSO based on the norm of dominant difference (MOPSO/DD) for dealing with MaOPs. Moreover, we design a L_p -norm-based density estimator which makes MOPSO/DD not only have good convergence and diversity but also have lower complexity. Experiments on benchmark problems demonstrate that our proposal is competitive with respect to the state-of-the-art MOPSOs and multiobjective evolutionary algorithms.

Software test data generation technology based on polymorphic particle swarm evolutionary algorithm

Combinatorial testing is a statute-based software testing method that aims to select a small number of valid test cases from a large combinatorial space of software under test to generate a set of test cases with high coverage and strong error debunking ability. However, combinatorial test case generation is an NP-hard problem that requires solving the combinatorial problem in polynomial time, so a meta-heuristic search algorithm is needed to solve the problem. Compared with other meta-heuristic search algorithms, the particle swarm algorithm is more competitive in terms of coverage table generation scale and execution time. In this paper, we systematically review and summarize the existing research results on generating combinatorial test case sets using particle swarm algorithm, and propose a combinatorial test case generation method that can handle arbitrary coverage strengths by combining the improved one-test-at-a-time strategy and the adaptive particle swarm algorithm for the variable strength combinatorial test problem and the parameter selection problem of the particle swarm algorithm. To address the parameter configuration problem of the particle swarm algorithm, the four parameters of inertia weight, learning factor, population size and iteration number are reasonably set, which makes the particle swarm algorithm more suitable for the generation of coverage tables. For the inertia weights.

Adaptively Allocating Constraint-Handling Techniques for Constrained Multi-objective Optimization Problems

For solving constrained multi-objective optimization problems (CMOPs), an effective constraint-handling technique (CHT) is of great importance. Recently, many CHTs have been proposed for solving CMOPs. However, no single CHT can outperform all kinds of CMOPs. This paper proposes an algorithm, namely, ACHT-M2M, which adaptively allocates the existing CHTs in an M2M framework for solving CMOPs. To be more specific, a CMOP is first decomposed into several constrained multi-objective optimization subproblems by ACHT-M2M. Each subproblem has a subpopulation in a subregion. CHT for each subregion is adaptively allocated according to a proposed composite performance measure. Population for the next generation is selected from subregions by selection operators with different CHTs and the obtained nondominated feasible solutions in each generation are used to update a predefined archive. ACHT-M2M assembles the advantages of different CHTs and makes them cooperate with each other. The proposed ACHT-M2M is finally compared with the other 12 representative algorithms on benchmark CMOPs and the experimental results further confirm the effectiveness of ACHT-M2M for solving CMOPs.

A Many-Objective Evolutionary Algorithm Based on a Two-Round Selection Strategy

Balancing population diversity and convergence is critical for evolutionary algorithms to solve many-objective optimization problems (MaOPs). In this paper, a two-round environmental selection strategy is proposed to pursue good tradeoff between population diversity and convergence for many-objective evolutionary algorithms (MaOEAs). Particularly, in the first round, the solutions with small neighborhood density are picked out to form a candidate pool, where the neighborhood density of a solution is calculated based on a novel adaptive position transformation strategy. In the second round, the best solution in terms of convergence is selected from the candidate pool and inserted into the next generation. The procedure is repeated until a new population is generated. The two-round selection strategy is embedded into an MaOEA framework and the resulting algorithm, namely, 2REA, is compared with eight state-of-the-art MaOEAs on various benchmark MaOPs. The experimental results show that 2REA is very competitive with the compared MaOEAs and the two-round selection strategy works well on balancing population diversity and convergence.

Two-stage multi-tasking transform framework for large-scale many-objective optimization problems

Real-world optimization applications in complex systems always contain multiple factors to be optimized, which can be formulated as multi-objective optimization problems. These problems have been solved by many evolutionary algorithms like MOEA/D, NSGA-III, and KnEA. However, when the numbers of decision variables and objectives increase, the computation costs of those mentioned algorithms will be unaffordable. To reduce such high computation cost on large-scale many-objective optimization problems, we proposed a two-stage framework. The first stage of the proposed algorithm combines with a multi-tasking optimization strategy and a bi-directional search strategy, where the original problem is reformulated as a multi-tasking optimization problem in the decision space to enhance the convergence. To improve the diversity, in the second stage, the proposed algorithm applies multi-tasking optimization to a number of sub-problems based on reference points in the objective space. In this paper, to show the effectiveness of the proposed algorithm, we test the algorithm on the DTLZ and LSMOP problems and compare it with existing algorithms, and it outperforms other compared algorithms in most cases and shows disadvantage on both convergence and diversity.

Evolutionary Algorithm for Knee-Based Multiple Criteria Decision Making

Although numerous effective and efficient multiobjective evolutionary algorithms have been developed in recent years to search for a well-converged and well-diversified Pareto optimal front, most of these designs are computationally expensive and have to maintain a large population of individuals throughout the evolutionary process. Once the Pareto optimal front is found satisfactorily, the cognitive burden is then imposed upon decision makers to handpick one solution for implementation among a massive number of candidates even with powerful multicriteria decision-making tools. With the increase in the number of decision variables and objective functions in the face of real-world applications, these problems have become a daunting challenge. In this article, we propose a recursive evolutionary algorithm, called EvoKnee^r, to directly search for global knee solutions, but also multiple local knee solutions using the minimum Manhattan distance approach as opposed to an enormous number of Pareto optimal solutions. Compared with the traditional evolutionary approaches, the proposed design herein only preserves nondominated solutions in rank one in each generation. Boundary Individuals Selection is tailored to select only M ² boundary individuals where M is the number of objectives. Relieving the burden of maintaining a large population size and its diversity throughout a lengthy evolutionary process, this design with a very low computational cost allows the evolutionary algorithm to converge to knee solutions quickly. To facilitate the experimental validations, a simulator with a graphical user interface is developed under the Delphi XE7 platform and made available for public use. In addition, the proposed algorithm is evaluated with the DO2DK, DEB2DK, DEB2DK2, and DEB3DK benchmark functions. The comparison results validate that the proposed EvoKnee^r algorithm is computationally and efficiently finding all global and local knee solutions.

A multi-objective hyper-heuristic algorithm based on adaptive epsilon-greedy selection

A variety of meta-heuristics have shown promising performance for solving multi-objective optimization problems (MOPs). However, existing meta-heuristics may have the best performance on particular MOPs, but may not perform well on the other MOPs. To improve the cross-domain ability, this paper presents a multi-objective hyper-heuristic algorithm based on adaptive epsilon-greedy selection (HH_EG) for solving MOPs. To select and combine low-level heuristics (LLHs) during the evolutionary procedure, this paper also proposes an adaptive epsilon-greedy selection strategy. The proposed hyper-heuristic can solve problems from varied domains by simply changing LLHs without redesigning the high-level strategy. Meanwhile, HH_EG does not need to tune parameters, and is easy to be integrated with various performance indicators. We test HH_EG on the classical DTLZ test suite, the IMOP test suite, the many-objective MaF test suite, and a test suite of a real-world multi-objective problem. Experimental results show the effectiveness of HH_EG in combining the advantages of each LLH and solving cross-domain problems.