Penalty-based Boundary Intersection Research Articles

An Improved NSGA-III with a Comprehensive Adaptive Penalty Scheme for Many-Objective Optimization

Pareto dominance-based algorithms face a significant challenge in handling many-objective optimization problems. As the number of objectives increases, the sharp rise in non-dominated individuals makes it challenging for the algorithm to differentiate their quality, resulting in a loss of selection pressure. The application of the penalty-based boundary intersection (PBI) method can balance convergence and diversity in algorithms. The PBI method guides the evolution of individuals by integrating the parallel and perpendicular distances between individuals and reference vectors, where the penalty factor is crucial for balancing these two distances and significantly affects algorithm performance. Therefore, a comprehensive adaptive penalty scheme was proposed and applied to NSGA-III, named caps-NSGA-III, to achieve balance and symmetry between convergence and diversity. Initially, each reference vector’s penalty factor is computed based on its own characteristic. Then, during the iteration process, the penalty factor is adaptively adjusted according to the evolutionary state of the individuals associated with the corresponding reference vector. Finally, a monitoring strategy is designed to oversee the penalty factor, ensuring that adaptive adjustments align with the algorithm’s needs at different stages. Through a comparison involving benchmark experiments and two real-world problems, the competitiveness of caps-NSGA-III was demonstrated.

Bi-level optimization and integrated multiple-criteria decision making under uncertainty for structural design of insulated bearings

To address the premature failure of insulated bearings, this article proposes a design method for the structure of bearings to prolong their service life. First, a bi-level optimization model is established to determine the structural parameters of bearings considering their insulating and mechanical properties. Upper-level problems include bearing voltage ratio, porosity of insulating coatings, minimum thickness of oil film and Hertz contact area. Lower-level problems include fatigue life of bearings and adhesion of insulating coatings. Second, a non-dominated sorting genetic algorithm-III based on space decomposition and penalty-based boundary intersection (NSGA-III-SD-PBI) is designed to solve the optimization model. Third, the integrated multiple-criteria decision-making (MCDM) using the fuzzy best–worst method (BWM) and fuzzy technique for order preference by similarity to an ideal solution (TOPSIS) is developed to determine the best structural parameters of bearings. A case study demonstrates the effectiveness and superiority of the proposed method, with fatigue life increased by 20%.

A new proximity metric based on optimality conditions for single and multi-objective optimization: Method and validation

With the widespread application of Evolutionary Algorithms (EAs), their performance needs to be evaluated using more than the usual performance metrics. In the EA literature, various metrics assess the convergence ability of these algorithms. However, many of them require prior knowledge of the Pareto-optimal front. Recently, two Karush–Kuhn–Tucker Proximity Metrics (KKTPMs) have been introduced to measure convergence without needing prior knowledge. One relies on the Augmented Achievement Scalarization Function (AASF) method (AASF-KKTPM), and the other on Benson’s method (B-KKTPM). However, both require specific parameters and reference points, making them computationally expensive. In this paper, we introduce a novel version of KKTPM applicable to single-, multi-, and many-objective optimization problems, utilizing the Penalty-based Boundary Intersection (PBI) method (PBI-KKTPM). Additionally, we introduce an approximate approach to reduce the computational burden of solving PBI-KKTPM optimization problems. Through extensive computational experiments across 23 case studies, our proposed metric demonstrates a significant reduction in computational cost, ranging from 20.68% to 60.03% compared to the computational overhead associated with the AASF-KKTPM metric, and from 16.48% to 61.15% compared to the computational overhead associated with the B-KKTPM metric. Noteworthy features of the proposed metric include its independence from knowledge of the true Pareto-optimal front and its applicability as a termination criterion for EAs. Another feature of the proposed metric is its ability to deal with black box problems very efficiently.

Use of Two Penalty Values in Multiobjective Evolutionary Algorithm Based on Decomposition.

The multiobjective evolutionary algorithm based on decomposition (MOEA/D) with the penalty-based boundary intersection (PBI) function (denoted as MOEA/D-PBI) has been frequently used in many studies in the literature. One essential issue in MOEA/D-PBI is its penalty parameter value specification. However, it is not easy to specify the penalty parameter value appropriately. This is because MOEA/D-PBI shows different search behavior when the penalty parameter values are different. The PBI function with a small penalty parameter value is good for convergence. However, the PBI function with a large value of penalty parameter is needed to preserve the diversity and uniformity of solutions. Although some methods for adapting the penalty parameter value for each weight vector have been proposed, they usually lead to slow convergence. In this article, we propose the idea of using two different values of penalty parameter simultaneously in MOEA/D-PBI. Although the idea is simple, the proposed algorithm is able to utilize both the convergence ability of a small penalty parameter value and the diversification ability of a large penalty parameter value of the PBI function. Experimental results demonstrate that the proposed algorithm works well on a wide range of test problems.

Surrogate-assisted operator-repeated evolutionary algorithm for computationally expensive multi-objective problems

Surrogate-assisted strategies work differently in surrogate-assisted multi-objective evolutionary algorithms. How to select suitable surrogate-assisted strategies to balance global and local search and diversity of Pareto optimal solutions is usually difficult for efficient optimization of computationally expensive multi-objective problems, which only allow limited time costs in the optimization process. Therefore, to further improve the optimization efficiency for expensive multi-objective problems, a surrogate-assisted operator-repeated multi-objective evolutionary algorithm is proposed by reasonably integrating several surrogate-assisted strategies in this study. Specifically, the proposed algorithm is based on a operator-repeated offspring creation strategy, which can produce many diverse candidate offspring individuals and thus make the proposed algorithm search globally and efficiently. In addition, a novel infill criterion termed as reference-vector-guided surrogate-assisted penalty-based boundary intersection is proposed and combined with a common Kriging-based expected improvement matrix infill criterion for complementary prescreening over the candidate offspring individuals. These two infill criteria can make a good balance between global search and diversity of Pareto optimal solutions. In addition, to fasten convergence speed, an improved surrogate-based multi-objective local search method with minimum distance-angle sampling is also proposed and embedded in the proposed algorithm. Several benchmark problems with dimensions varying from 8 to 50 and a practical two-objective airfoil design optimization problem with 14 design variables are tested to validate efficiency of the proposed algorithm. The experimental results demonstrate that the proposed algorithm significantly outperforms some state-of-the-art surrogate-assisted multi-objective optimization algorithms on most of problems.

Identify potential circRNA-disease associations through a multi-objective evolutionary algorithm

More and more studies have demonstrated that circRNAs can be used as markers of various diseases due to their inherent stability. Numerous computational methods, especially artificial intelligence approaches, have been applied to the prediction of circRNA-disease associations. However, the objective functions of these prediction methods are usually single and standard. The single objective function hardly reflects the characteristics of the prediction problem, and leads to low prediction accuracy. There has never been a method to design a set of objective functions for the circRNA-disease prediction problem and solve it by using intelligent optimization algorithms. In this paper, a method ICDMOE is proposed to identify circRNA-disease associations via a multi-objective evolutionary algorithm. A solution space is defined and four objective functions are designed based on matrix factorization and modularity of similarity networks. An improved decomposition-based multi-objective evolutionary algorithm (MOEA/D) is also employed to solve these objective functions. In the MOEA/D, an adaptive penalty-based boundary intersection strategy is proposed, which not only ensures convergence, but also guarantees the diversity of solutions in the Pareto front. Finally, the experimental results show that ICDMOE has better performance than pure matrix factorization-based methods, non-adaptive MOEA/D-based methods and other prediction methods. Furthermore, we find that the predicted disease-circRNAs can be confirmed by existing studies, miRNA regulations and expression profiles, etc. These indicate that ICDMOE can provide good candidates for biomedical experiments.

Compensation and profit allocation for collaborative multicenter vehicle routing problems with time windows

With the growing interest in win–win cooperation within the logistics industry recently, collaborative multicenter logistics networks have witnessed the establishment of numerous alliances aimed at achieving cost savings. However, alliance members may choose to withdraw from an alliance due to changes in market positioning strategies and short-term profit considerations. Such withdraws often lead to the disintegration of the alliance and result in financial losses for the remaining non-defaulting members. This study focuses on addressing the issues of compensation and profit allocation in a specific scenario known as the collaborative multicenter vehicle routing problem with time windows and defaulting members withdrawal (CMVRPTWDMW). The objective is to ensure alliance stability and fair profit distribution among its members. To achieve this, a tri-objective mathematical model is proposed to optimize the re-formed alliances by considering the total operating cost, number of vehicles, and service waiting time. Additionally, an improved minimum cost remaining saving method is introduced to accurately quantify the deserved profits of non-defaulting members. Compensation quantification and default cost allocation models are then developed based on this analysis. To solve the CMVRPTWDMW and obtain Pareto optimal solutions, a hybrid heuristic algorithm is designed. This algorithm combines a 3D k-means clustering algorithm for customer reassignment after the withdrawal of defaulting members, effectively reducing the complexity of the problem. Furthermore, the algorithm employs a Clarke-Wright saving method-based non-dominated sorting genetic algorithm-III (CW-NSGA-III) to search for optimal solutions. To enhance the algorithm’s search ability, an elite alteration strategy and a penalty-based boundary intersection approach are incorporated into CW-NSGA-III. Finally, a real-world case study conducted in Chongqing City, China, is presented to demonstrate the feasibility of the proposed models and solution algorithms. The study contributes valuable insights into the sustainable operation of urban logistics networks that may experience unpredictable defaults, providing guidance for the stability of logistics alliances in the transportation sector.

Mathematical foundation, discussion and suggestion on penalty parameter setting of penalty-based boundary intersection method for many-objective optimization problems

Mathematical foundation, discussion and suggestion on penalty parameter setting of penalty-based boundary intersection method for many-objective optimization problems

Resilient Optimal Defensive Strategy of TSK Fuzzy-Model-Based Microgrids' System via a Novel Reinforcement Learning Approach.

With consideration of false data injection (FDI) on the demand side, it brings a great challenge for the optimal defensive strategy with the security issue, voltage stability, power flow, and economic cost indexes. This article proposes a Takagi-Sugeuo-Kang (TSK) fuzzy system-based reinforcement learning approach for the resilient optimal defensive strategy of interconnected microgrids. Due to FDI uncertainty of the system load, TSK-based deep deterministic policy gradient (DDPG) is proposed to learn the actor network and the critic network, where multiple indexes' assessment occurs in the critic network, and the security switching control strategy is made in the actor network. Alternating direction method of multipliers (ADMM) method is improved for policy gradient with online coordination between the actor network and the critic network learning, and its convergence and optimality are proved properly. On the basis of security switching control strategy, the penalty-based boundary intersection (PBI)-based multiobjective optimization method is utilized to solve economic cost and emission issues simultaneously with considering voltage stability and rate-of-change of frequency (RoCoF) limits. According to simulation results, it reveals that the proposed resilient optimal defensive strategy can be a viable and promising alternative for tackling uncertain attack problems on interconnected microgrids.

PBI Based Multi-Objective Optimization via Deep Reinforcement Elite Learning Strategy for Micro-Grid Dispatch With Frequency Dynamics

Micro-grid dispatch is facing a great challenge since the system security and economic cost are often time-varying with different requirements. In this paper, a novel multiple time-scale dispatch model is firstly constructed with considering load voltage and frequency dynamics of islanding awareness, which is limited with rate-of-change of frequency (RoCoF) of micro-grid. Based on this model, micro-grid power dispatch is converted into a multi-objective optimization problem (MOP) with taking economic cost, voltage deviation and frequency stability into consideration. To address this problem, a penalty-based boundary intersection (PBI) based multi-objective optimization approach is developed with an elite learning technique. In the above approach, a developed deep deterministic policy gradient (DDPG) is proposed to learn evolutionary parameters with grid based alternating weight vectors, which can lead to better convergence ability and diversity distribution. Simulation results show that the proposed optimization strategy can properly achieve minimum economic cost with simultaneously satisfying voltage and frequency stability, which provides a viable and promising way for tackling with micro-grid dispatch problem.

A collaborative decomposition-based evolutionary algorithm integrating normal and penalty-based boundary intersection methods for many-objective optimization

Decomposition-based evolutionary algorithms have recently become fairly popular for many-objective optimization owing to their excellent selection pressure. However, existing decomposition methods are still quite sensitive to the various shapes of the frontiers of many-objective optimization problems (MaOPs). On the one hand, the penalty-based boundary intersection (PBI) method is incapable of acquiring uniform frontiers for MaOPs with very convex frontiers due to the radial distribution of the reference lines. On the other hand, the parallel reference lines of the normal boundary intersection (NBI) method often result in poor diversity for MaOPs with concave frontiers because of under-sampling near the boundaries. In this paper, a collaborative decomposition (CoD) method is first proposed to integrate the advantages of the PBI and NBI methods to overcome their respective disadvantages. This method inherits the NBI-style Tchebycheff function as a convergence measure to improve the convergence and uniformity of the distribution of the PBI method. Moreover, this method also adaptively tunes the extent to which an NBI reference line is rotated towards a PBI reference line for every boundary subproblem to enhance the diversity of the distribution of the NBI method. Furthermore, a CoD-based evolutionary algorithm (CoDEA) is presented for many-objective optimization. A CoD-based ranking strategy is primarily designed in the CoDEA to rank all the individuals associated with every boundary subproblem according to the CoD aggregation function and determine the best ranks. The proposed algorithm is compared with several popular many-objective evolutionary algorithms on 85 benchmark test instances. The experimental results show that the CoDEA achieves high competitiveness, benefiting from the CoD method.

Real-time and Coordinated UAV Path Planning for Road Traffic Surveillance: A Penalty-based Boundary Intersection Approach

Real-time and Coordinated UAV Path Planning for Road Traffic Surveillance: A Penalty-based Boundary Intersection Approach

A Multi-Objective Quantum-Inspired Seagull Optimization Algorithm Based on Decomposition for Unmanned Aerial Vehicle Path Planning

The traditional seagull optimization algorithm cannot handle multi-objective optimization problems, so a multi-objective quantum-inspired seagull optimization algorithm based on decomposition (MOQSOA/D) is proposed. Multi-objective computing and quantum computing are introduced into MOQSOA/D. MOQSOA/D transforms the multi-objective problem into multiple scalar optimization sub-problems, and establishes a dynamic archive and a leadership archive at the same time. The Pareto solution of each sub-problem is stored in a dynamic archive, and the non-dominated Pareto solution is stored in the leader archive. While processing each sub-problem, each seagull is represented by a string of qubits, which is used to calculate the current seagull direction of flight, and a variable angular-distance rotation (VAR) gate is used to change the probability amplitude of the qubits, thereby updating the direction of flight. Penalty-based boundary intersection approach is introduced to determine whether the generated Pareto solution is retained. The proposed algorithm and six different algorithms were tested on 69 indicators, and the results show that the algorithm achieved better results in 40 indicators. In addition, a Unmanned Aerial Vehicle (UAV) path planning model in three-dimensional environment is designed to test the utility of MOQSOA/D, and the algorithm is compared with the other algorithm to demonstrate its effectiveness.

DMO-QPSO: A Multi-Objective Quantum-Behaved Particle Swarm Optimization Algorithm Based on Decomposition with Diversity Control

The decomposition-based multi-objective evolutionary algorithm (MOEA/D) has shown remarkable effectiveness in solving multi-objective problems (MOPs). In this paper, we integrate the quantum-behaved particle swarm optimization (QPSO) algorithm with the MOEA/D framework in order to make the QPSO be able to solve MOPs effectively, with the advantage of the QPSO being fully used. We also employ a diversity controlling mechanism to avoid the premature convergence especially at the later stage of the search process, and thus further improve the performance of our proposed algorithm. In addition, we introduce a number of nondominated solutions to generate the global best for guiding other particles in the swarm. Experiments are conducted to compare the proposed algorithm, DMO-QPSO, with four multi-objective particle swarm optimization algorithms and one multi-objective evolutionary algorithm on 15 test functions, including both bi-objective and tri-objective problems. The results show that the performance of the proposed DMO-QPSO is better than other five algorithms in solving most of these test problems. Moreover, we further study the impact of two different decomposition approaches, i.e., the penalty-based boundary intersection (PBI) and Tchebycheff (TCH) approaches, as well as the polynomial mutation operator on the algorithmic performance of DMO-QPSO.

Diversity preference-based many-objective particle swarm optimization using reference-lines-based framework

A simple yet effective diversity preference approach is developed and coupled with many-objective particle swarm optimization (PSO) for solving box-constraint optimization problems. Since the selection of global leaders in PSO is crucial for many-objective optimization, an approach is developed using the reference-lines-based framework to update these leaders. In the proposed approach, diversity is ensured first by making clusters of solutions for every reference line. Thereafter, only one solution from each cluster with minimum penalty-based boundary intersection (PBI) fitness is selected. In case any cluster of a line is empty, a solution with minimum PBI fitness with respect to the line is selected. This ensures the selection of isolated but sometimes dominated solutions. The proposed approach is then used for updating the archive of global leaders and for assigning them to particles in the swarm. The proposed algorithm, which is referred to as MaOPSO-DP, is tested on 3-, 5-, 8-, 10-, and 15-objective instances of DTLZ and WFG test problems. Results demonstrate the effectiveness of MaOPSO-DP over eight many-objective evolutionary and PSO algorithms from the literature.

MOEA/D-Based Probabilistic PBI Approach for Risk-Based Optimal Operation of Hybrid Energy System With Intermittent Power Uncertainty

The stochastic nature of intermittent energy resources has brought significant challenges to the optimal operation of the hybrid energy systems. This article proposes a probabilistic multiobjective evolutionary algorithm based on decomposition (MOEA/D) method with two-step risk-based decision-making strategy to tackle this problem. A scenario-based technique is first utilized to generate a stochastic model of the hybrid energy system. Those scenarios divide the feasible domain into several regions. Then, based on the MOEA/D framework, a probabilistic penalty-based boundary intersection (PBI) with gradient descent differential evolution (GDDE) algorithm is proposed to search the optimal scheme from these regions under different uncertainty budgets. To ensure reliable and low risk operation of the hybrid energy system, the Markov inequality is employed to deduce a proper interval of the uncertainty budget. Further, a fuzzy grid technique is proposed to choose the best scheme for real-world applications. The experimental results confirm that the probabilistic adjustable parameters can properly control the uncertainty budget and lower the risk probability. Further, it is also shown that the proposed MOEA/D-GDDE can significantly enhance the optimization efficiency.

An Improved Penalty-Based Boundary Intersection Approach for Irregular Problems

The multi-objective evolutionary algorithm based on decomposition (MOEA/D) has achieved remarkable success in regular optimizing multi-objective problems. In practice, there are lots of irregular problems. The projection of their Pareto fronts (PF) may not cover the whole triangle on the first octant. There is an empty PF (EPF) or complex PF area. These problems significantly reduce the performance of MOEA/D. In this paper, an improved algorithm is proposed to deal with such complex problems. The basic idea is: (1) the PBI method is used to ensure the convergence and simultaneously keep diversity over the triangle; (2) a non-dominated and maximum distance based selection is used to replace the dominated objective individuals selected by the PBI method. It is referred to selection based on the PBI and Non-dominated Maximum distance (MOEAD-PNM). First, the mathematical foundation between the penalty factor and contour line is deduced. Based on the foundation, the penalty factor setting method is proposed and the PBI-based method is used to select an individual for each reference vector. The reference vectors directing to the EPF induce “bad” solutions. Then, these solutions are gradually replaced by better individuals selected according to the non-dominated and maximum distance-based selection strategy. The proposed two-step selection method can guarantee the wideness and uniformity of the “good” solution set. Finally, the performance of the MOEAD-PNM algorithm and five classic algorithms on more than ten test problems are compared. The experimental results show the competitiveness and effectiveness of the proposed algorithm on these challenging problems.

Research on hot rolling scheduling problem based on Two-phase Pareto algorithm.

Under the background of excess capacity and energy saving in iron and steel enterprises, the hot rolling batch scheduling problem based on energy saving is a multi-objective and multi constraint optimization problem. In this paper, a hybrid multi-objective prize-collecting vehicle routing problem (Hybrid Price Collect Vehicle Routing Problem, HPCVRP) model is established to ensure minimum energy consumption, meet process rules, and maximize resource utilization. A two-phase Pareto search algorithm (2PPLS) is designed to solve this model. The improved MOEA/D with a penalty based boundary intersection distance (PBI) algorithm (MOEA/D-PBI) is introduced to decompose the HPCVRP in the first phase. In the second phase, the multi-objective ant colony system (MOACS) and Pareto local search (PLS) algorithm is used to generate approximate Pareto-optimal solutions. The final solution is then selected according to the actual demand and preference. In the simulation experiment, the 2PPLS is compared with five other algorithms, which shows the superiority of 2PPLS. Finally, the experiment was carried out on actual slab data from a steel plant in Shanghai. The results show that the model and algorithm can effectively reduce the energy consumption in the process of hot rolling batch scheduling.

A many-objective evolutionary algorithm based on decomposition with dynamic resource allocation for irregular optimization

The multi-objective optimization problem has been encountered in numerous fields such as high-speed train head shape design, overlapping community detection, power dispatch, and unmanned aerial vehicle formation. To address such issues, current approaches focus mainly on problems with regular Pareto front rather than solving the irregular Pareto front. Considering this situation, we propose a many-objective evolutionary algorithm based on decomposition with dynamic resource allocation (MaOEA/D-DRA) for irregular optimization. The proposed algorithm can dynamically allocate computing resources to different search areas according to different shapes of the problem’s Pareto front. An evolutionary population and an external archive are used in the search process, and information extracted from the external archive is used to guide the evolutionary population to different search regions. The evolutionary population evolves with the Tchebycheff approach to decompose a problem into several subproblems, and all the subproblems are optimized in a collaborative manner. The external archive is updated with the method of shift-based density estimation. The proposed algorithm is compared with five state-of-the-art many-objective evolutionary algorithms using a variety of test problems with irregular Pareto front. Experimental results show that the proposed algorithm out-performs these five algorithms with respect to convergence speed and diversity of population members. By comparison with the weighted-sum approach and penalty-based boundary intersection approach, there is an improvement in performance after integration of the Tchebycheff approach into the proposed algorithm.

Multi-objective decomposition optimization algorithm based on adaptive weight vector and matching strategy

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) uses pre-set weight vector and random matching mechanism between sub-problems and individuals, which makes the algorithm simple and efficient. However, when solving the problem of discontinuous Pareto Front, this will lead to not only the decline of the diversity of population, but also the degradation of the performance of solution set. To solve these problems, a multi-objective decomposition optimization algorithm based on adaptive weight vector and matching strategy (MOEA/D-AVM) is proposed in the article. Firstly, the algorithm finds the invalid sub-problems in the discontinuous region, and then updates these sub-problems according to the current evolutionary stage. It can reduce the possibility that the invalid sub-problems mislead evolutionary process. Secondly, the matching mechanism is established according to the value of penalty-based boundary intersection (PBI) and the Euclidean distance between the sub-problems and individuals. This mechanism can enhance the relationship between individuals and sub-problems. Finally, individuals who perform well in the neighborhood replacement operation are saved in the external archive. It can improve the diversity of the optimal solution set obtained by the algorithm. The proposed algorithm is compared with other related algorithms in the standard test problem. The result shows that the solution set obtained by MOEA/D-AVM not only can better cover the Pareto Front, but also has a competitive performance in solving the problem of discontinuous Pareto Front.