Difficulties of the NSGA-II with the Many-Objective LeadingOnes Problem

TL;DR

It is shown that the NSGA-II, with a population size by at most a constant factor larger than the Pareto front, cannot compute the Pareto front in less than exponential time.

Abstract

The NSGA-II is the most prominent multi-objective evolutionary algorithm (cited more than 50,000 times). Very recently, a mathematical runtime analysis has proven that this algorithm can have enormous difficulties when the number of objectives is larger than two (Zheng, Doerr. IEEE Transactions on Evolutionary Computation (2024)). However, this result was shown only for the OneMinMax benchmark problem, which has the particularity that all solutions are on the Pareto front, a fact heavily exploited in the proof of this result. In this work, we show a comparable result for the LeadingOnesTrailingZeroes benchmark. This popular benchmark problem appears more natural in that most of its solutions are not on the Pareto front. With a careful analysis of the population dynamics of the NGSA-II optimizing this benchmark, we manage to show that when the population grows on the Pareto front, then it does so much faster by creating known Pareto optima than by spreading out on the Pareto front. Consequently, already when still a constant fraction of the Pareto front is unexplored, the crowding distance becomes the crucial selection mechanism, and thus the same problems arise as in the optimization of OneMinMax. With these and some further arguments, we show that the NSGA-II, with a population size by at most a constant factor larger than the Pareto front, cannot compute the Pareto front in less than exponential time.

Figures (4)

• Figure 1: The number of distinct objective values on the Pareto front of the NSGA-II (\ref{['nsga']}) with standard bit mutation and fair selection for the combined parent and offspring population ($R_t$) and the selected population ($P_{t+1}$) over $1000$ iterations. The algorithm is run on the $4$-LOTZ problem of size $n = 40$, with the dashed line showing the total size of the Pareto front ($M = 441$) for reference. From top to bottom, each pair of curves refers, respectively, to the population size $N = 8M$, $N = 4M$, and $N = 2M$.
• Figure 2: The number of individuals in the non-dominated subpopulation $F_1$ as well as the number of Pareto-optimal individuals in the combined parent and offspring population ($R_t$) and the selected population ($P_{t + 1}$) of the NSGA-II (\ref{['nsga']}) with standard bit mutation and fair selection over $1000$ iterations. The algorithm is run on the $4$-LOTZ problem of size $n = 40$, resulting in a total size of the Pareto front of $M = 441$, with $N = 4M$. The dashed lines show the total population size of $R_t$ (namely $2N$) and of $P_{t + 1}$ (namely $N$) for reference. This is the same run as in \ref{['fig:pareto_fronts_all']} for $N = 4M$.
• Figure 3: The number of distinct objective values on the Pareto front of the NSGA-II (\ref{['nsga']}) with standard bit mutation and different selection methods for the combined parent and offspring population ($R_t$) over $1000$ iterations. The algorithm is run on the $4$-LOTZ problem of size $n = 40$, with the dashed line showing the total size of the Pareto front ($M = 441$) for reference. The population size of the algorithm is $N = 4M$. The data for the run of the Standard algorithm is the same as in \ref{['fig:pareto_fronts_all']} with $N = 4M$.
• Figure 4: The number of distinct objective values on the Pareto front of the NSGA-II (\ref{['nsga']}) with one-bit mutation and fair selection for the combined parent and offspring population ($R_t$) and the selected population ($P_{t+1}$) over $1000$ iterations. The algorithm is run on the $4$-LOTZ problem of size $n = 40$, with the dashed line showing the total size of the Pareto front ($M = 441$) for reference. From top to bottom, each pair of curves refers, respectively, to the population size $N = 8M$, $N = 4M$, and $N = 2M$.

Theorems & Definitions (11)

• Theorem 1
• Lemma 2
• Lemma 3: Random selection lemma
• proof
• Lemma 4
• Lemma 5
• proof
• proof : Proof of \ref{['lm2']}
• proof : Proof of \ref{['thm:main']}
• Theorem 6