# Improve Pareto Dominance relation

The Pareto dominance relation, as the most commonly adopted ranking method, plays an essential role in multi-objective evolutionary algorithms (MOEAs). However, its ability is often severely degraded with the increase of the number of objectives [1]. The major reason for this performance deterioration is that individuals are not likely to be dominated by others. Given a point in m-dimensional objective space, any ε-ball of a point can be partitioned into the incomparable, the dominated and dominating region. The ratio between the size of the incomparable region, and the dominated (and dominating) region decreases proportionally to 1/2m-1.

Recently, some researchers have proposed the use of relaxed forms of Pareto dominance as a way of regulating convergence of MOEAs, such as, ⍺-dominance [2], 𝜖-dominance [3], cone 𝜖-dominance [4], CDAS [5], GrEA [6] and angle dominance criterion [7]. Under these relaxed definitions, a solution has a higher chance to be dominated by other solutions and the selection pressure toward the Pareto front is increased.

In an MOEA, if a solution can dominate more area based on the adopted dominance relation, the algorithm is capable of exploring more solutions and hence accelerating convergence. To this end, we suggest enhancing the Pareto dominance order from the geometric perspective. To be specific, the Pareto order cone is widened by rotating the edges of the standard Pareto order cone around the origin towards the outside, and convex obtuse cones which can dominate a larger area are generated.

The figure above shows how the dominance relation has been changed when the edge-rotated cone order is applied. In the left image, y and y′ are mutually non-dominated by each other because neither of them is in the dominating space of the other point. However, when an edge-rotated cone is adopted in the right image, the point y′ is dominated by y. We can see that the edge-rotated cones provide a stricter order compared to the Pareto order. They can guide the search towards the Pareto front better as they establish an ordering among the incomparable solutions (with respect to the Pareto order) in the sense that better incomparable solutions are preferred.

For the detailed implementation of the edge-rotated cone and the algorithm of using it in MOEAs, especially its performance compared to the Pareto dominance relation, please refer to [8].

**References:**

[1] Khare, V., Yao, X., Deb, K.: Performance scaling of multi-objective evolutionary algorithms. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Thiele, L., Deb, K. (eds.) EMO 2003. LNCS, vol. 2632, pp. 376–390. Springer, Heidelberg (2003).

[2] Ikeda, K., Kita, H., Kobayashi, S.: Failure of Pareto-based MOEAs: does non- dominated really mean near to optimal? In: Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No. 01TH8546), vol. 2, pp. 957–962. IEEE, May 2001

[3] Laumanns, M., Thiele, L., Deb, K., Zitzler, E.: Combining convergence and diver- sity in evolutionary multiobjective optimization. Evol. Comput. 10(3), 263–282 (2002)

[4] Batista, L.S., Campelo, F., Guimar ̃aes, F.G., Ram ́ırez, J.A.: Pareto cone epsilon- dominance: improving convergence and diversity in multiobjective evolutionary algorithms. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds.) EMO 2011. LNCS, vol. 6576, pp. 76–90. Springer, Heidelberg (2011).

[5] Sato, H., Aguirre, H., Tanaka, K.: Controlling dominance area of solutions in multi- objective evolutionary algorithms and performance analysis on multiobjective 0/1 knapsack problems. IPSJ Digital Courier 3, 703–718 (2007)

[6] Yang, S., Li, M., Liu, X., Zheng, J.: A grid-based evolutionary algorithm for many- objective optimization. IEEE Trans. Evol. Comput. 17(5), 721–736 (2013)

[7] Liu, Y., Zhu, N., Li, K., Li, M., Zheng, J., Li, K.: An angle dominance criterion for evolutionary many-objective optimization. Inf. Sci. 509, 376–399 (2020)

[8] Wang, Y., Deutz, A, Ba ̈ck, T. and Emmerich, M., 2020, September. Improving Many-Objective Evolutionary Algorithms by Means of Edge- Rotated Cones. In International Conference on Parallel Problem Solving from Nature (in press). Springer.