Evolutionary algorithms based on hypervolume have demonstrated good performance for solving many-objective optimization problems. However, hypervolume needs prohibitively expensive computational effort. This paper proposes a simplified hypervolume calculation method which can be used to roughly evaluate the convergence and diversity of solutions. The main idea is to use the nearest neighbors of a particular solution to calculate the volume as the solution’s hypervolume value. Moreover, this paper improves the selection operator and the update strategy of external population according to the simplified hypervolume. Then, the proposed algorithm (SHEA) is compared with some state-of-the-art algorithms on fifteen test functions of CEC2018 MaOP competition, and the experimental results prove the feasibility of the proposed algorithm.

Multiobjective optimization problems (MOPs) have been applied in numerous real-world applications. A minimized MOP which often has two or three objectives can be defined as follows [

In the last few decades, multiobjective evolutionary algorithms (MOEAs) [

To solve these problems, there are three main categories of many-objective evolutionary algorithms (MaOEAs). The first category is based on the modified dominance relationship [

The second category uses decomposition-based method to solve MaOPs. The main idea is to decompose a many-objective optimization problem into a set of subproblems and optimize them collaboratively. The most representative algorithms are MOEA/

The third approach is indicator-based evolutionary algorithms. Indicators such as hypervolume [

So the key question is how to reduce the computational complexity and keep the advantages of the hypervolume indicator at the same time. The major contributions of this paper can be summarized as follows.

A simplified hypervolume calculation method is proposed to roughly evaluate the convergence and diversity of solutions

To enhance the quality of offspring, a new selection operator based on the simplified hypervolume is proposed to choose excellent parents

The simplified hypervolume together with nondomination is used in the new update strategy of external population to store solutions with good convergence and diversity

In the remainder of this paper, Section

A simplified hypervolume-based evolutionary algorithm for many-objective optimization (SHEA) is proposed to solve MaOPs. The core part of this paper is a new hypervolume calculation method to roughly evaluate the convergence and diversity of solutions. Furthermore, this new hypervolume is used to improve selection operator and update strategy.

To get the hypervolume value, the normalized population

Thereafter, the volume between the particular solution and the reference point is calculated. As for the boundary solutions, just calculate the volumes between these boundary solutions and its adjacent solutions and remove the terms when the objective function value of

Then, the minimum of the particular solution’s volumes for all objectives is kept as the hypervolume value (

When

The new selection operator aims to choose parents with good convergence and diversity to generate high quality offspring in differential evolution [

The new selection operator calculates the nondominated neighbors’ hypervolume and retains the top

Then, randomly choose

The external population is adopted to store solutions with good convergence and diversity. The solutions with smaller hypervolume values of the nondominated solutions of population are deleted to maintain the number of the external population. And to retain diversity,

SHEA works as follows (Algorithm

MaOP(1)

A stopping criterion

To update the population, the solutions in set

Otherwise, find the solution with the minimum Tchebycheff function in

In the proposed algorithm,

In this section, the proposed algorithm is compared with four state-of-the-art algorithms such as NSGAIII [

All of these fifteen test problems for each algorithm mentioned above are run on PlatEMO [

Comparison experiment employs inverted generational distance (IGD) [

IGD can comprehensively measure the convergence and diversity of population, and when IGD value is smaller, the population is closer to Pareto fronts. For each problem, around 10000 points on Pareto fronts are uniformly sampled to calculate IGD. Besides, in the sense of statistics comparison experiment, use Wilcoxon rank-sum test [

Table

The mean and standard deviation values of IGD obtained by SHEA, NSGAIII, MOEA/DD, KnEA, and RVEA while “+,” “=,” “−” mean SHEA is better than, the same as, and worse than the compared algorithms.

Problem | SHEA | NSGAIII | MOEA/DD | KnEA | RVEA |
---|---|---|---|---|---|

MaF1-5 | 2.0830 | 3.0206 | 1.3136 | 3.2908 | |

MaF2-5 | 1.3012 | 1.3665 | 1.3734 | 1.2736 | |

MaF3-5 | 9.7048 | 1.1693 | 1.6971 | 8.0906 | |

MaF4-5 | 3.2220 | 7.7080 | 2.9005 | 4.8145 | |

MaF5-5 | 2.5845 | 6.1823 | 2.6408 | 2.3218 | |

MaF6-5 | 4.9054 | 7.6075 | 8.0721 | 9.6534 | |

MaF7-5 | 3.4413 | 1.7891 | 3.2757 | 4.4844 | |

MaF8-5 | 2.1547 | 3.3063 | 2.9874 | 4.8799 | |

MaF9-5 | 6.5706 | 2.5294 | 5.6348 | 3.6742 | |

MaF10-5 | 5.9695 | 5.4591 | 5.1174 | 4.3054 | |

MaF11-5 | 9.5446 | 4.6337 | 5.7805 | 5.7122 | |

MaF12-5 | 1.1183 | 1.2858 | 1.2609 | 1.1224 | |

MaF13-5 | 2.9677 | 2.4087 | 2.2221 | 6.6957 | |

MaF14-5 | 5.7391 | 7.9085 | 7.6615 | 7.1001 | |

MaF15-5 | 8.3177 | 1.0511 | 3.4169 | 6.1117 | |

MaF1-10 | 3.0459 | 3.1563 | 4.8879 | 6.6423 | |

MaF2-10 | 1.9990 | 2.3692 | 2.9196 | 4.8292 | |

MaF3-10 | 9.3337 | 1.1204 | 1.1840 | 9.8006 | |

MaF4-10 | 9.3360 | 1.2678 | 4.3124 | 2.3329 | |

MaF5-10 | 1.1938 | 2.9174 | 8.1774 | 1.0827 | |

MaF6-10 | 2.1879 | 1.2060 | 8.4648 | 3.3999 | |

MaF7-10 | 9.6508 | 1.1620 | 2.7026 | 2.4904 | |

MaF8-10 | 4.6344 | 9.1389 | 2.6509 | 1.0244 | |

MaF9-10 | 8.9161 | 9.5771 | 8.5718 | 1.1161 | |

MaF10-10 | 1.2266 | 1.2608 | 1.2117 | 1.1948 | |

MaF11-10 | 4.2556 | 1.4289 | 1.3490 | 1.3839 | |

MaF12-10 | 5.1072 | 6.0813 | 5.2874 | 4.8913 | |

MaF13-10 | 4.1263 | 4.4820 | 2.1228 | 9.3878 | |

MaF14-10 | 8.7548 | 1.4523 | 1.7663 | 6.6978 | |

MaF15-10 | 2.0029 | 3.0468 | 1.9364 | 1.0594 | |

MaF1-15 | 4.1181 | 6.3920 | 3.3929 | 7.3558 | |

MaF2-15 | 2.2440 | 2.4645 | 3.1534 | 7.8173 | |

MaF3-15 | 1.0297 | 1.7607 | 1.1719 | 2.2295 | |

MaF4-15 | 1.8657 | 4.5055 | 1.5432 | 7.7257 | |

MaF5-15 | 2.2259 | 3.1331 | 7.3038 | 3.2985 | |

MaF6-15 | 3.7141 | 1.6153 | 4.8429 | 1.9771 | |

MaF7-15 | 7.7037 | 3.3764 | 2.4545 | 4.3845 | |

MaF8-15 | 4.0978 | 1.5460 | 1.9336 | 1.1960 | |

MaF9-15 | 2.2160 | 1.3164 | 5.2669 | 1.6125 | |

MaF10-15 | 1.7186 | 1.6348 | 1.9986 | 1.7417 | |

MaF11-15 | 8.8230 | 1.8586 | 2.1955 | 1.9469 | |

MaF12-15 | 8.4176 | 8.8410 | 1.1457 | 9.1349 | |

MaF13-15 | 1.8693 | 3.8155 | 6.1376 | 1.3296 | |

MaF14-15 | 2.1776 | 1.4120 | 4.2956 | 7.8507 | |

MaF15-15 | 4.3682 | 7.6508 | 1.1636 | 1.4233 | |

+/−/= | — | 32/9/4 | 29/12/4 | 25/11/9 | 30/10/5 |

The bold values indicate best performance.

On all forty-five problems in Table

In fifteen MaFs, there are 8 problems (F1, F2, F4, F5, F7, F8, F9, and F15) that have partial PFs. The PF projections of these problems do not fully cover the unit hyperplane. The mean IGDs of SHEA are smaller than those of NSGAIII, MOEA/DD, KnEA, and RVEA on twenty-two, nineteen, sixteen, and twenty problems, separately. And for 6 problems (F3, F10, F11, F12, F13, and F14) with PF projection fully covering the unit hyperplane, there are respectively eleven, eleven, twelve, twelve problems that the mean IGDs of SHEA are smaller than those of NSGAIII, MOEA/DD, KnEA, and RVEA. As for the problem F6 whose PF is degraded, SHEA is superior to NSGAIII, MOEA/DD, KnEA, and RVEA on three problems in the form of IGD. All of these comparison results mentioned above indicate the best overall performance of SHEA on most problems and prove the excellent performance of simplified hypervolume in estimating convergence and diversity.

To simplify the calculation of hypervolume, a new simplified hypervolume is proposed to roughly estimate the convergence and diversity of solutions; then the new method is used in the selection operator and the update strategy of external population. And the proposed algorithm indicates good performance according to comparing experimental results with four state-of-the-art algorithms.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (nos. 61806120, 61502290, 61401263, 61672334, and 61673251), China Postdoctoral Science Foundation (no. 2015M582606), Industrial Research Project of Science and Technology in Shaanxi Province (nos. 2015GY016 and 2017JQ6063), Fundamental Research Fund for the Central Universities (no. GK202003071), and Natural Science Basic Research Plan in Shaanxi Province of China (no. 2016JQ6045).