SETNDS: A SET-Based Non-Dominated Sorting Algorithm for Multi-Objective Optimization Problems

Non-dominated sorting, used to find pareto solutions or assign solutions to different fronts, is a key but time-consuming process in multi-objective evolutionary algorithms (MOEAs). The best-case and worst-case time complexity of non-dominated sorting algorithms currently known are O(MNlogN) and O(MN2); M and N represent the number of objectives and the population size, respectively. In this paper, a more efficient SET-based non-dominated sorting algorithm, shorted to SETNDS, is proposed. The proposed algorithm can greatly reduce the number of comparisons on the promise of ensuring a shorter running time. In SETNDS, the rank of a solution to be sorted is determined by only comparing with the one with the highest rank degree in its dominant set. This algorithm is compared with six generally existing non-dominated sorting algorithms—fast non-dominated sorting, the arena’s principle sort, the deductive sort, the corner sort, the efficient non-dominated sort and the best order sort on several kinds of datasets. The compared results show that the proposed algorithm is feasible and effective and its computational efficiency outperforms other existing algorithms.

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An Efficient Multi-Objective Robust Optimization Method by Sequentially Searching From Nominal Pareto Solutions

Journal of Computing and Information Science in Engineering ◽

10.1115/1.4049996 ◽

2021 ◽

Vol 21 (4) ◽

Author(s):

Tingting Xia ◽

Mian Li

Keyword(s):

Robust Optimization ◽

Pareto Front ◽

Optimization Problem ◽

Optimization Problems ◽

Computational Time ◽

Pareto Solutions ◽

Worst Case ◽

Multi Objective ◽

Pareto Points ◽

Single Objective

Abstract Multi-objective optimization problems (MOOPs) with uncertainties are common in engineering design. To find robust Pareto fronts, multi-objective robust optimization (MORO) methods with inner–outer optimization structures usually have high computational complexity, which is a critical issue. Generally, in design problems, robust Pareto solutions lie somewhere closer to nominal Pareto points compared with randomly initialized points. The searching process for robust solutions could be more efficient if starting from nominal Pareto points. We propose a new method sequentially approaching to the robust Pareto front (SARPF) from the nominal Pareto points where MOOPs with uncertainties are solved in two stages. The deterministic optimization problem and robustness metric optimization are solved in the first stage, where nominal Pareto solutions and the robust-most solutions are identified, respectively. In the second stage, a new single-objective robust optimization problem is formulated to find the robust Pareto solutions starting from the nominal Pareto points in the region between the nominal Pareto front and robust-most points. The proposed SARPF method can reduce a significant amount of computational time since the optimization process can be performed in parallel at each stage. Vertex estimation is also applied to approximate the worst-case uncertain parameter values, which can reduce computational efforts further. The global solvers, NSGA-II for multi-objective cases and genetic algorithm (GA) for single-objective cases, are used in corresponding optimization processes. Three examples with the comparison with results from the previous method are presented to demonstrate the applicability and efficiency of the proposed method.

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An Efficient Multi-Objective Robust Optimization Method by Sequentially Searching From Nominal Pareto Solutions

Volume 11B: 46th Design Automation Conference (DAC) ◽

10.1115/detc2020-22155 ◽

2020 ◽

Author(s):

Tingting Xia ◽

Mian Li

Keyword(s):

Robust Optimization ◽

Pareto Front ◽

Optimization Problem ◽

Optimization Problems ◽

Pareto Solutions ◽

Worst Case ◽

Multi Objective ◽

Pareto Points ◽

Engineering Design Problems ◽

Single Objective

Abstract Multi-objective optimization problems (MOOPs) with uncertainties are common in engineering design problems. To find the robust Pareto fronts, multi-objective robust optimization methods with inner-outer optimization structures generally have high computational complexity, which is always an important issue to address. Based on the general experience, robust Pareto solutions usually lie somewhere near the nominal Pareto points. Starting from the obtained nominal Pareto points, the search process for robust solutions could be more efficient. In this paper, we propose a method that sequentially approaching to the robust Pareto front (SARPF) from the nominal Pareto points. MOOPs are solved by the SARPF in two optimization stages. The deterministic optimization problem and the robustness metric optimization problem are solved in the first stage, and nominal Pareto solutions and the robust-most solutions can be found respectively. In the second stage, a new single-objective robust optimization problem is formulated to find the robust Pareto solutions starting from the nominal Pareto points in the region between the nominal Pareto front and the robust-most points. The proposed SARPF method can save a significant amount of computation time since the optimization process can be performed in parallel at each stage. Vertex estimation is also applied to approximate the worst-case uncertain parameter values which can save computational efforts further. The global solvers, NSGA-II for the multi-objective case and genetic algorithm (GA) for the single-objective case, are used in corresponding optimization processes. Two examples with comparison to a previous method are presented for the applicability and efficiency demonstration.

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A Constrained Multi-Objective Evolutionary Algorithm Based on Boundary Search and Archive

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001416590023 ◽

2015 ◽

Vol 30 (01) ◽

pp. 1659002 ◽

Cited By ~ 10

Author(s):

Hai-Lin Liu ◽

Chaoda Peng ◽

Fangqing Gu ◽

Jiechang Wen

Keyword(s):

Evolutionary Algorithm ◽

Optimization Problems ◽

Search Method ◽

Constraint Handling ◽

Multi Objective Optimization ◽

Pareto Solutions ◽

Genetic Operator ◽

Multi Objective ◽

Leading Role

In this paper, we propose a decomposition-based evolutionary algorithm with boundary search and archive for constrained multi-objective optimization problems (CMOPs), named CM2M. It decomposes a CMOP into a number of optimization subproblems and optimizes them simultaneously. Moreover, a novel constraint handling scheme based on the boundary search and archive is proposed. Each subproblem has one archive, including a subpopulation and a temporary register. Those individuals with better objective values and lower constraint violations are recorded in the subpopulation, while the temporary register consists of those individuals ever found before. To improve the efficiency of the algorithm, the boundary search method is designed. This method makes the feasible individuals with a higher probability to perform genetic operator with the infeasible individuals. Especially, when the constraints are active at the Pareto solutions, it can play its leading role. Compared with two algorithms, i.e. CMOEA/D-DE-CDP and Gary’s algorithm, on 18 CMOPs, the results show the effectiveness of the proposed constraint handling scheme.

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Search Acceleration of Evolutionary Multi-Objective Optimization Using an Estimated Convergence Point

Mathematics ◽

10.3390/math7020129 ◽

2019 ◽

Vol 7 (2) ◽

pp. 129 ◽

Cited By ~ 4

Author(s):

Yan Pei ◽

Jun Yu ◽

Hideyuki Takagi

Keyword(s):

Parameter Space ◽

Statistical Tests ◽

Distribution Characteristic ◽

Multi Objective Optimization ◽

Pareto Improvement ◽

Pareto Solutions ◽

Pareto Solution ◽

Multi Objective ◽

Objective Space ◽

Convergence Point

We propose a method to accelerate evolutionary multi-objective optimization (EMO) search using an estimated convergence point. Pareto improvement from the last generation to the current generation supports information of promising Pareto solution areas in both an objective space and a parameter space. We use this information to construct a set of moving vectors and estimate a non-dominated Pareto point from these moving vectors. In this work, we attempt to use different methods for constructing moving vectors, and use the convergence point estimated by using the moving vectors to accelerate EMO search. From our evaluation results, we found that the landscape of Pareto improvement has a uni-modal distribution characteristic in an objective space, and has a multi-modal distribution characteristic in a parameter space. Our proposed method can enhance EMO search when the landscape of Pareto improvement has a uni-modal distribution characteristic in a parameter space, and by chance also does that when landscape of Pareto improvement has a multi-modal distribution characteristic in a parameter space. The proposed methods can not only obtain more Pareto solutions compared with the conventional non-dominant sorting genetic algorithm (NSGA)-II algorithm, but can also increase the diversity of Pareto solutions. This indicates that our proposed method can enhance the search capability of EMO in both Pareto dominance and solution diversity. We also found that the method of constructing moving vectors is a primary issue for the success of our proposed method. We analyze and discuss this method with several evaluation metrics and statistical tests. The proposed method has potential to enhance EMO embedding deterministic learning methods in stochastic optimization algorithms.

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