An Opposition-Based Evolutionary Algorithm for Many-Objective Optimization with Adaptive Clustering Mechanism

Abstract

Balancing convergence and diversity has become a key point especially in many-objective optimization where the large numbers of objectives pose many challenges to the evolutionary algorithms. In this paper, an opposition-based evolutionary algorithm with the adaptive clustering mechanism is proposed for solving the complex optimization problem. In particular, opposition-based learning is integrated in the proposed algorithm to initialize the solution, and the nondominated sorting scheme with a new adaptive clustering mechanism is adopted in the environmental selection phase to ensure both convergence and diversity. The proposed method is compared with other nine evolutionary algorithms on a number of test problems with up to fifteen objectives, which verify the best performance of the proposed algorithm. Also, the algorithm is applied to a variety of multiobjective engineering optimization problems. The experimental results have shown the competitiveness and effectiveness of our proposed algorithm in solving challenging real-world problems.

Figure 1

An illustration of the approach to generate the reference point and the reference vector in three objective spaces. As the figure shows, given the H₁=2 and H₂=1, respectively, the final number of the reference point is 9.

Figure 2

An example showing the initialization phase in OBEA.

Figure 3

An illustration of the acute angle θ, distance du, and distance dp.

Figure 4

Parallel coordinates of the nondominated front obtained by each algorithm on 15-objective WG5 in the run associated with the median HV value. (a) OBEA on WFG5. (b) RVEA on WFG5. (c) NSGA-III on WFG5. (d) MOEA/DD on WFG5. (e) SPEAR on WFG5. (f) MOEA/DVA on WFG5. (g) Two Arch2 on WFG5.

Figure 5

Ranking and score of average performance obtained by each compared algorithms in terms of HV.

Figure 6

(a) Average performance score obtained by eleven algorithms over all test problems of different numbers of objectives in terms of the HV and (b) average performance score obtained by ten algorithms on dimensions for different test problems in terms of the HV, Dx for DTLZ, and Wx for WFG. The values of the proposed OBEA are connected by a solid red line.

Figure 7

(a) An illustration of the special situation when solely using the acute angle or perpendicular distance to select individual. (b) An illustration of the three functions versus the iterative generations.

Figure 8

(a) An illustration of the IGD results on 15-objective DTLZ2 when using different μ. (b) An illustration of the IGD results on 10-objective WFG7 when using different μ. (c) An illustration of the IGD results on 8-objective DTLZ4 when using different μ. (d) An illustration of the IGD results on 5-objective WFG4 when using different μ.

Figure 9

Trajectory of the mean IGD value on ten algorithms with fifteen objectives. (a) DTLZ1. (b) DTLZ4. (c) DTLZ7. (d) WFG4. (e) WFG7. (f) WFG9.

Algorithm 1

General framework of OBEA.

Algorithm 2

Environmental selection.

Algorithm 3

Adaptive clustering operation.

Algorithm 4

Opposition-based selection.