Penalty-based Boundary Intersection Research Articles

Decomposition-based hyperheuristic approaches for the bi-objective cold chain considering environmental effects

This paper proposed a novel approach for a practical version of the cold chain, namely location-routing problem-based low-carbon cold chain (LRPLCCC). In the proposed bi-objective model, the first objective is the total logistics cost, including the fixed costs of the opened depots and leased vehicles, as well as the cost of fuel consumption and carbon emissions, and the second is to minimize the amount of quality degradation that aims at improving clients’ satisfaction and maintain product freshness. The cargos of clients are classified into three types: general, refrigerated, and frozen cargos. Since the presented problem is NP-hard, a novel multi-objective hyperheuristic (MOHH) was proposed to obtain the Pareto solutions. In this framework, three selection strategies were developed to improve the performance of MOHH, that is, random simple, choice function, and FRR-MAB (fitness rate rank based multi-armed bandit), and three acceptance criteria using the decomposition approaches in MOEA/D were also developed, namely penalty-based boundary intersection, Tchebycheff, and modified Tchebycheff approaches. Extensive experiments were provided to verify the efficiency of the proposed algorithms and assessed the effects of algorithm parameters on the Pareto front. The results showed that the efficiency of the proposed algorithm outperforms several existing well-known multi-objective evolutionary algorithms (MOEA).

A survey of decomposition approaches in multiobjective evolutionary algorithms

Since the multiobjective evolutionary algorithm based on decomposition (MOEA/D) was proposed by Zhang and Li in 2007, this interesting framework has attracted a considerable attention from researchers. In MOEA/D, a multiobjective optimization problem is decomposed into a series of aggregated subproblems, which are optimized simultaneously in a collaborative way by using the information from their neighboring subproblems. The decomposition approach has significant impact on MOEA/D as it directs the evolutionary search. Many improved MOEA/D variants proposed various kinds of decomposition approaches and have shown promising performance for different kinds of problems. In this paper, we give a survey of decomposition approaches, which are classified into five categories, i.e., the tradition decomposition, the modified Tchebycheff decomposition, the modified penalty-based boundary intersection decomposition, the constrained decomposition, and other special cases of decomposition. Moreover, discussions are further given in this paper to analyze the performance of different decomposition approaches. One clarifies the difference between Tchebycheff decomposition and Pareto-based domination. The other one compares the performance of various decomposition approaches on different benchmark problems. Experiments results have demonstrated that the Tchebycheff decomposition and its varieties are robust on solving most problems while some specific decomposition approaches are very effective for some problems with special features.

Research on Many-Objective Flexible Job Shop Intelligent Scheduling Problem Based on Improved NSGA-III

With the development of intelligent manufacturing and the customized product demand of customers, manufacturing enterprises are urgently required to carry out high-efficiency, high-quality, flexibility and low-cost manufacturing to enhance the competitiveness of enterprises. Intelligent job shop scheduling problem is the core decision of intelligent manufacturing production management. Many-objective job shop scheduling algorithms can effectively solve this problem. However, existing optimization algorithms cannot effectively solve many-objective flexible job shop scheduling problem. This paper establishes the many-objective job shop intelligent scheduling model with complex constraints, and proposes an improved intelligent decision optimization algorithm named NSGA-III-APEV based on NSGA-III. This algorithm uses the penalty-based boundary intersection distance that takes into account both convergence and diversity simultaneously to define the distance between the population individual and the reference vector in the association operation. This paper exploits the penalty-based boundary intersection distance-based elimination mechanism to preserve individuals and reduce the computational cost in the individual preservation strategy. Meanwhile, the adaptive mutation strategy based on consanguinity is employed in genetic operators. The presented method effectively improves the convergence and diversity of the population. Finally, NSGA-III-APEV with other algorithms was compared through benchmarks. Experimental results demonstrated the effectiveness and superiority of the improved method. The feasibility of the improved method in solving the many-objective flexible job shop scheduling problem are verified by engineering examples.

Efficient network architecture search via multiobjective particle swarm optimization based on decomposition

The efforts devoted to manually increasing the width and depth of convolutional neural network (CNN) usually require a large amount of time and expertise. It has stimulated a rising demand of neural architecture search (NAS) over these years. However, most popular NAS approaches solely optimize for low prediction error without penalizing high structure complexity. To this end, this paper proposes MOPSO/D-Net, a CNN architecture search method with multiobjective particle swarm optimization based on decomposition (MOPSO/D). The main goal is to reformulate NAS as a multiobjective evolutionary optimization problem, where the optimal architecture is learned by minimizing two conflicting objectives, namely the error rate of classification and number of parameters of the network. Along with the hybrid binary encoding and adaptive penalty-based boundary intersection, an improved MOPSO/D is further proposed to solve the formulated multiobjective NAS and provide diverse tradeoff solutions. Experimental studies verify the effectiveness of MOPSO/D-Net compared with current manual and automated CNN generation methods. The proposed algorithm achieves impressive classification performance with a small number of parameters on each of two benchmark datasets, particularly, 0.4% error rate with 0.16M params on MNIST and 5.88% error rate with 8.1M params on CIFAR-10, respectively.

PBI function based evolutionary algorithm with precise penalty parameter for unconstrained many-objective optimization

Fixed or experiential penalty parameter of the penalty-based boundary intersection (PBI) function method cannot simultaneously ensure the convergence and diversity for all shape of Pareto front (PF). Too large penalty parameter may lead to bad convergence while too small parameter can not ensure the diversity. Specially, if the penalty parameter is too small, some reference weight vectors may have no solution on it. This error is hard to be rectified. In this paper, we prove that the lower bound of the penalty parameter is determined by three factors. The first one is the shape of the PF. The second one is the cosine distance between two adjacent reference vectors. The third one is the number of objectives. We deduce the lower bound of the penalty parameter. Once the penalty parameter was calculated, an individual with minimal PBI function is attached to the corresponding reference vector. The minimal-PBI-function-first principle is used in the environmental selection to guarantee the wideness and uniformity of the solution set. The time complexity is low. The proposed method is compared with other three state-of-the-art many-objective evolutionary algorithms on the unconstrained test problems MaOP, DTLZ and WFG with up to fifteen objectives. The experimental results show the competitiveness and effectiveness of the proposed algorithm in both time efficiency and accuracy.

The Collaborative Local Search Based on Dynamic-Constrained Decomposition With Grids for Combinatorial Multiobjective Optimization.

The decomposition-based algorithms [e.g., multiobjective evolutionary algorithm based on decomposition (MOEA/D)] transform a multiobjective optimization problem (MOP) into a number of single-objective optimization subproblems and solve them in a collaborative manner. It is a natural framework for using single-objective local search (LS) to solve combinatorial MOPs. However, commonly used decomposition methods, such as weighted sum (WS), Tchebycheff (TCH), and penalty-based boundary intersection (PBI) may not be good at maintaining the population diversity while providing diverse initial solutions for different LS procedures in a collaborative way. Based on our previous work on the constrained decomposition with grids (CDG), this article proposes a dynamic CDG (DCDG) framework used to design a multiobjective memetic algorithm (DCDG-MOMA). DCDG uses grids for maintaining diversity, supporting the collaborative LS. In addition, DCDG dynamically increases the number of grids for obtaining more nondominated solutions as well as the better collaborative search among them. DCDG-MOMA has been compared with several classical and state-of-the-art algorithms on multiobjective traveling salesman problem (MOTSP), multiobjective quadratic assignment problem (MOQAP), and multiobjective capacitated arc routing problem (MOCARP).

Improved NSGA-III with selection-and-elimination operator

Abstract A fast non-dominated sorting genetic algorithm based on reference-point strategy (NSGA-III) is a well-known many-objective optimization algorithm in which the reference-point strategy is incorporated to maintain population diversity. However, the convergence capability of NSGA-III is poor in many cases. In this paper, a new selection-and-elimination operator is designed to balance convergence and diversity. First, a selection operator is employed to identify the reference point with the minimum niche count, and then one individual with the shortest penalty-based boundary intersection distance is chosen. Second, a reference point with the maximum niche count is identified, and one individual with the longest penalty-based boundary intersection distance is removed by the elimination operator. To test performance, this modification is verified on benchmark problems with up to 15 objectives, and compared with five other state-of-the-art algorithms. Simulation results demonstrate that our modified strategy can achieve the better performance.

An adaptive penalty-based boundary intersection method for many-objective optimization problem

Compared with domination-based methods, the multi-objective evolutionary algorithm based on decomposition (MOEA/D) is less prone to the difficulty caused by an increase in the number of objectives. It is a promising algorithmic framework for solving many-objective optimization problems (MaOPs). In MOEA/D, the target MaOP is decomposed into a set of single-objective problems by using a scalarizing function with evenly specified weight vectors. Among the available scalarizing functions, penalty-based boundary intersection (PBI) with an appropriate penalty parameter is known to perform well. However, its performance is heavily influenced by the setting of the penalty factor (θ), which can take a value from zero to +∞. A limited amount of work has thus far considered the choice of an appropriate value of θ. This paper presents a comprehensive experimental study on WFG and WFG-extend problems featuring two to 15 objectives. A range of values of θ is investigated to understand its influence on the performance of the PBI-based MOEA/D (MOEA/D-PBI). Based on the observations, the range of values of θ are divided into three sub-regions, and a two-stage adaptive penalty scheme is proposed to adaptively choose an appropriate value from 0.001 to 8000 during an optimization run. The results of experiments show that, the robustness of MOEA/D-PBI can be significantly enhanced using the proposed scheme.

An Improved NSGA-III Algorithm Using Genetic K-Means Clustering Algorithm

The non-dominated sorting genetic algorithm III (NSGA-III) has recently been proposed to solve many-objective optimization problems (MaOPs). While this algorithm achieves good diversity, its convergence is unsatisfactory. In order to improve the convergence, we propose an improved NSGA-III using a genetic K-means clustering algorithm (NSGA-III-GKM), which can also ensure diversity and automatically provide the number and direction vector of the subspaces. Compared with the NSGA-III, the proposed NSGA-III-GKM has two key features. First, the initial reference points are clustered using a GKM clustering algorithm, which realizes automatic learning of the number of clusters. Second, as the reference points are replaced by cluster centers, a penalty-based boundary intersection (PBI) aggregation function is introduced to replace the perpendicular distance. The proposed NSGA-III-GKM and other similar optimization algorithms (NSGA-III, MOEA/D, U-NSGA-III, DC-NSGA-III and B-NSGA-III) are tested on DTLZ test problems and UF test problems. The simulation results demonstrate that the NSGA-III-GKM exhibits better diversity and convergence performance than the other algorithms.

Optimization of an Intelligent Community Hybrid Energy System With Robustness Consideration

An intelligent community hybrid energy system (ICHES) includes renewable energy power generation equipment, distributed controllable generation equipment, energy storage systems, mobile agents, and large numbers of fixed loads. In recent years, many studies of hybrid energy systems have been considered from the perspective of cost. However, the pollution and robustness of the system cannot be ignored. To address these issues, this paper introduces the environmental impact ratio of power generation and power supply robustness into the model and proposes a novel intelligent community hybrid energy optimization model. The proposed model is a multi-objective optimization problem. To address this complex multi-objective optimization problem, a direction vector adjustment (DVA) mechanism is introduced into the multi-objective evolutionary algorithm based on decomposition (MOEA/D) using localized penalty-based boundary intersection (LPBI) (MOEA/D-LPBI). Then, an improved MOEA/D-LPBI-DVA is proposed. The experimental results show that our model is more reasonable and outperforms existing models. The solutions obtained allow the problem to have a better effect, thereby effectively optimizing the hybrid energy output and achieving the multi-objective optimization requirements of low cost, low pollution, and high robustness of the system power supply.

A decomposition-based multiobjective evolutionary algorithm with angle-based adaptive penalty

A multiobjective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multiobjective optimization problem (MOP) into a number of scalar optimization subproblems and optimizes them in a collaborative manner. In MOEA/D, decomposition mechanisms are used to push the population to approach the Pareto optimal front (POF), while a set of uniformly distributed weight vectors are applied to maintain the diversity of the population. Penalty-based boundary intersection (PBI) is one of the approaches used frequently in decomposition. In PBI, the penalty factor plays a crucial role in balancing convergence and diversity. However, the traditional PBI approach adopts a fixed penalty value, which will significantly degrade the performance of MOEA/D on some MOPs with complicated POFs. This paper proposes an angle-based adaptive penalty (AAP) scheme for MOEA/D, called MOEA/D-AAP, which can dynamically adjust the penalty value for each weight vector during the evolutionary process. Six newly designed benchmark MOPs and an MOP in the wastewater treatment process are used to test the effectiveness of the proposed MOEA/D-AAP. Comparison experiments demonstrate that the AAP scheme can significantly improve the performance of MOEA/D.

A Grid Weighted Sum Pareto Local Search for Combinatorial Multi and Many-Objective Optimization.

Combinatorial multiobjective optimization problems (CMOPs) are very popular due to their widespread applications in the real world. One common method for CMOPs is Pareto local search (PLS), a natural extension of single-objective local search (LS). However, classical PLS tends to reserve all of the nondominated solutions for LS, which causes the inefficient LS, as well as unbearable computational and space cost. Due to the aforementioned reasons, most PLS approaches can only handle CMOPs with no more than two objectives. In this paper, by combining the Pareto dominance and weighted sum (WS) approach in a grid system, the grid weighted sum dominance (gws-dominance) is proposed and integrated into PLS for CMOPs with multiple objectives. In the grid system, at most one representative solution is maintained in each grid for more efficient LS, thus largely reducing the computational and space complexity. The grid-based WS approach can further guide the LS in different grids for maintaining more widely and uniformly distributed Pareto front approximations. In the experimental studies, the grid WS PLS is compared with the classical PLS, three decomposition-based LS approaches [multiobjective evolutionary algorithm based on decomposition-LS (WS, Tchebycheff, and penalty-based boundary intersection)], a grid-based algorithm ( ϵ -MOEA), and a state-of-the-art hybrid approach (multiobjective memetic algorithm based on decomposition) on two sets of benchmark CMOPs. The experimental results show that the grid weighted sum Pareto local search significantly outperforms the compared algorithms and remains effective and efficient on combinatorial multiobjective and even many-objective optimization problems.

A New Decomposition-Based NSGA-II for Many-Objective Optimization

Multiobjective evolutionary algorithms (MOEAs) have proven their effectiveness and efficiency in solving problems with two or three objectives. However, recent studies show that MOEAs face many difficulties when tackling problems involving a larger number of objectives as their behavior becomes similar to a random walk in the search space since most individuals are nondominated with respect to each other. Motivated by the interesting results of decomposition-based approaches and preference-based ones, we propose in this paper a new decomposition-based dominance relation to deal with many-objective optimization problems and a new diversity factor based on the penalty-based boundary intersection method. Our reference point-based dominance (RP-dominance), has the ability to create a strict partial order on the set of nondominated solutions using a set of well-distributed reference points. The RP-dominance is subsequently used to substitute the Pareto dominance in nondominated sorting genetic algorithm-II (NSGA-II). The augmented MOEA, labeled as RP-dominance-based NSGA-II, has been statistically demonstrated to provide competitive and oftentimes better results when compared against four recently proposed decomposition-based MOEAs on commonly-used benchmark problems involving up to 20 objectives. In addition, the efficacy of the algorithm on a realistic water management problem is showcased.

An analysis of control parameters of MOEA/D under two different optimization scenarios

An unbounded external archive (UEA), which stores all nondominated solutions found during the search process, is frequently used to evaluate the performance of multi-objective evolutionary algorithms (MOEAs) in recent studies. A recent benchmarking study also shows that decomposition-based MOEA (MOEA/D) is competitive with state-of-the-art MOEAs when the UEA is incorporated into MOEA/D. However, a parameter study of MOEA/D using the UEA has not yet been performed. Thus, it is unclear how control parameter settings influence the performance of MOEA/D with the UEA. In this paper, we present an analysis of control parameters of MOEA/D under two performance evaluation scenarios. One is a final population scenario where the performance assessment of MOEAs is performed based on all nondominated solutions in the final population, and the other is a reduced UEA scenario where it is based on a pre-specified number of selected nondominated solutions from the UEA. Control parameters of MOEA/D investigated in this paper include the population size, scalarizing functions, and the penalty parameter of the penalty-based boundary intersection (PBI) function. Experimental results indicate that suitable settings of the three control parameters significantly depend on the choice of an optimization scenario. We also analyze the reason why the best parameter setting is totally different for each scenario.

A Scalar Projection and Angle-Based Evolutionary Algorithm for Many-Objective Optimization Problems.

In decomposition-based multiobjective evolutionary algorithms, the setting of search directions (or weight vectors), and the choice of reference points (i.e., the ideal point or the nadir point) in scalarizing functions, are of great importance to the performance of the algorithms. This paper proposes a new decomposition-based many-objective optimizer by simultaneously using adaptive search directions and two reference points. For each parent, binary search directions are constructed by using its objective vector and the two reference points. Each individual is simultaneously evaluated on two fitness functions-which are motivated by scalar projections-that are deduced to be the differences between two penalty-based boundary intersection (PBI) functions, and two inverted PBI functions, respectively. Solutions with the best value on each fitness function are emphasized. Moreover, an angle-based elimination procedure is adopted to select diversified solutions for the next generation. The use of adaptive search directions aims at effectively handling problems with irregular Pareto-optimal fronts, and the philosophy of using the ideal and nadir points simultaneously is to take advantages of the complementary effects of the two points when handling problems with either concave or convex fronts. The performance of the proposed algorithm is compared with seven state-of-the-art multi-/many-objective evolutionary algorithms on 32 test problems with up to 15 objectives. It is shown by the experimental results that the proposed algorithm is flexible when handling problems with different types of Pareto-optimal fronts, obtaining promising results regarding both the quality of the returned solution set and the efficiency of the new algorithm.

A modified PBI approach for multi-objective optimization with complex Pareto fronts

Abstract The penalty-based boundary intersection (PBI) approach is widely used in the decomposition-based multi-objective evolutionary algorithm (MOEA/D). Generally, a uniform distribution of weight vectors in PBI approach will lead to a set of evenly distributed solutions on the Pareto-optimal front (POF), but this approach cannot work well in practice when the target multi-objective optimization problem (MOP) has a complex POF. For example, the POF may have disconnected regions and a long tail and a sharp peak and a degenerate geometry, which significantly degrades the performance of the original MOEA/D. This paper proposes a modified PBI (MPBI) approach and a strategy of adjusting reference points (ARP) to handle these MOPs with complex fronts. A two-stage strategy is adopted in the proposed algorithm. The first stage is to determine a hyperplane based on the modified PBI approach, so that the projection points derived from the solutions obtained in second stage to this hyperplane are all in the first quadrant. Exploring those regions where the solution exists is also a key task in this stage. The second stage is to adjust the reference points periodically so that the reference points can be redistributed adaptively to improve the distribution of solutions. The framework of the proposed algorithm is based on θ -DEA and named NSGA-MPBI. Some widely used test instances and three many-objective MOPs with complex POFs are employed in the experiments. The experimental results indicate that NSGA-MPBI outperforms the state-of-the-art algorithms.

Enhancing robustness of the inverted PBI scalarizing method in MOEA/D

The scalarizing function design is an important issue influencing significantly the performance of a decomposition based multi-objective optimization algorithm (MOEA/D). Recently, an inverted penalty-based boundary intersection (IPBI) scalarizing function was proposed to improve the spread of solutions obtained by MOEA/D. Despite its effectiveness, MOEA/D with IPBI scalarizing function (MOEA/D-IPBI) still has several shortcomings: MOEA/D-IPBI often fails to obtain any solution within certain Pareto front (PF) regions. Furthermore, it may produce and retain unwanted dominated solutions outside the PF for some problems. In this work, we first analyze the reasons for the above two shortcomings of the IPBI scalarizing function, and then propose two improvement strategies, i.e., the adaptive reference point setting strategy and the adaptive subproblem replacement strategy, to overcome the two shortcomings of the IPBI scalarizing function respectively, giving rise to an enhanced MOEA/D with robust IPBI scalarizing method (R-IPBI). Experimental studies on WFG benchmark problems and the real-world reservoir flood control operation problems suggest that the two improvement strategies are very effective in overcoming the two shortcomings of the IPBI scalarizing function. As a result, the proposed R-IPBI algorithm is shown to be able to outperform the original MOEA/D-IPBI reliably.

Expected Improvement of Penalty-Based Boundary Intersection for Expensive Multiobjective Optimization

Computationally expensive multiobjective optimization problems are difficult to solve using solely evolutionary algorithms (EAs) and require surrogate models, such as the Kriging model. To solve such problems efficiently, we propose infill criteria for appropriately selecting multiple additional sample points for updating the Kriging model. These criteria correspond to the expected improvement of the penalty-based boundary intersection (PBI) and the inverted PBI. These PBI-based measures are increasingly applied to EAs due to their ability to explore better nondominated solutions than those that are obtained by the Tchebycheff function. In order to add sample points uniformly in the multiobjective space, we assign territories and niche counts to uniformly distributed weight vectors for evaluating the proposed criteria. We investigate these criteria in various test problems and compare them with established infill criteria for multiobjective surrogate-based optimization. Both proposed criteria yield better diversity and convergence than those obtained with other criteria for most of the test problems.

An improved biogeography/complex algorithm based on decomposition for many-objective optimization

It is difficult to maintain the balance between convergence and diversity for many-objective optimization problems (MaOPs) in the algorithms of evolutionary multi-objective (EMO). EMO algorithms are useful technology to solve the multi-objective optimization problems (MOPs). However, with the larger of optimization objectives, enough Pareto selection pressure will be loosed, and results in the performance of the algorithms are significantly reduced. The decomposition-based EMO developed for MaOPs have been shown to be effective, and the BBO algorithm is a low-complexity algorithm. In this paper, a hybrid decomposition-based BBO/Complex algorithm (HDB/BBO) for MaOPs is proposed. First, a set of uniformly distributed weight vectors and K-means aggregate method is introduced for decomposing MaOPs into several subsystems. Then, inferior migrated islands will not be chosen unless they pass the Metropolis criterion twice during the within-subsystem migration and cross-subsystem migration. The penalty-based boundary intersection (PBI) distance to calculate neighbor islands distance for balancing the algorithm of convergence and diversity. Finally, after mutation and clear duplication, a uniform distribution Pareto set can be obtained. Experimental results on both DTLZ and WFG benchmarks problems demonstrate the superiority of the proposed algorithm in comparison with three state-of-the-art algorithms in terms of both convergence and diversity.

An improved NSGA-III algorithm based on elimination operator for many-objective optimization

Due to the low selection pressure of the Pareto-dominance relation and the ineffectivity of diversity maintenance schemes in the environmental selection, the classical Pareto-dominance based multi-objective evolutionary algorithms (MOEAs) fail to handle many-objective optimization problems. The recently presented non-dominated sorting genetic algorithm III (NSGA-III) employs the uniformly distributed reference points to significantly promote population diversity, but the convergence based on the Pareto-dominance relation could still be enhanced. For this purpose, an improved NSGA-III algorithm based on elimination operator (NSGA-III-EO) is proposed. In the proposed algorithm, the elimination operator first identifies the reference point with maximum niche count and then employs the penalty-based boundary intersection distance to rank the individuals associated with it. To this end, the selection scheme is used to remove the worse individuals rather than to select the superior individuals. The proposed NSGA-III-EO is tested on a number of well-known benchmark problems with up to fifteen objectives and shows the competitive performance compared with five state-of-the-art MOEAs. Additionally, it is also tested on constrained problems having a large number of objectives and shows good performance.