NSGA-II: Balanced Multi-objective Genetic Algorithm

• NSGA-II is a multi-objective evolutionary algorithm that approximates Pareto fronts via fast non-dominated sorting and crowding distance mechanisms.
• It integrates a rarity-based tie-breaking rule to promote diverse, rare objective values and improve convergence in high-dimensional search spaces.
• The balanced selection approach offers provable polynomial runtime improvements over classic NSGA-II, mitigating sensitivity to population size and exponential delays in many-objective setups.

The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is the most widely adopted multi-objective evolutionary algorithm (MOEA) for approximating Pareto fronts in discrete and continuous search spaces. NSGA-II employs fast non-dominated sorting to organize the population into Pareto fronts, followed by a crowding distance mechanism to ensure diversity and uniform spread. However, recent mathematical analyses have detected fundamental limitations in the standard NSGA-II, particularly in high-dimensional (many-objective) settings and in its pronounced sensitivity to population size on classical combinatorial benchmarks. The "Balanced NSGA-II" introduces a rarity-based tie-breaking rule that provably circumvents both these pathologies, enabling polynomial-time convergence in settings where classic NSGA-II fails.

1. The Classic NSGA-II Workflow and Its Limitations

Standard NSGA-II maintains a population of fixed size $N$. In each generation:

• Variation: Each individual produces one offspring via standard bit mutation (each bit flipped independently with probability $1/n$) or with a crossover operator, depending on problem domain and implementation.
• Population Union: Parents and offspring are merged, forming $R = P\cup Q$ of size $2N$.
• Non-dominated Sorting: $R$ is decomposed into fronts $\mathcal{F}_1,\mathcal{F}_2,\dots$ by Pareto rank.
• Survivor Selection: Next parent generation $P'$ is filled with entire Pareto fronts in order until adding the next would overflow $N$; any overflow in the final partial front is resolved by descending crowding distance.
• Crowding Distance: For each front $F$, the crowding distance of individual $x$ is the sum, over objectives $j$, of normalized gaps between its immediate neighbors in $F$ sorted by $f_j$; boundaries receive infinite distance to preserve extremal solutions.

Critically, inside the partially accepted front $F_{j^*}$, any ties in crowding distance are resolved uniformly at random. This leads to loss of rare objective vectors and poor coverage of the Pareto front when $|F_{j^*}| \gg N$ and many solutions have identical (usually zero) crowding distance. Mathematical analyses rigorously establish two deficiencies in this scheme (Doerr et al., 16 Dec 2024):

• Many-objective (m > 2) Hardness: For classic NSGA-II, even with $N$ linear in the Pareto front size, the runtime to cover all Pareto points is exponential in $n^{\lceil m/2\rceil}$ for $m \geq 3$ (Doerr et al., 16 Dec 2024).
• Population-size Sensitivity: In the bi-objective regime, runtime scales at least linearly with $N$ (e.g., $\Theta(N n^k)$ on the OneJumpZeroJump problem), so "oversizing" the population significantly degrades performance (Doerr et al., 16 Dec 2024).

2. The Balanced Tie-breaking Rule: Rarity-Based Survivor Selection

The sole modification in balanced NSGA-II is the survivor selection among equally ranked and equally crowded individuals in the critical (overflowed) front. The algorithm replaces uniform-at-random tie-breaking with a rarity-aware group sampling procedure:

1. Grouping: Partition the tied set $C$ (with $s$ individuals to select) by unique objective vector values, yielding groups $C'_\ell$ for each distinct value $u_\ell$.
2. Quotas: For $\ell = 1,\dots, a$ (number of distinct objective vectors), select at most $\lfloor s/a\rfloor$ individuals uniformly at random from each $C'_\ell$. The union $W$ accumulates these picks.
3. Fill-up: If $|W| < s$, assign any remaining slots by uniform random draws from $C \setminus W$.

This rarity-based selection ensures every surviving front value is retained as evenly as possible, promoting rare objective vectors and preventing their elimination by the pancaking effect of purely random tie-breaking.

Pseudocode for the balanced tie-breaker:

C = critical_front_after_rank_and_crowding groups = partition C by objective value W = [] for group in groups: t = min(len(group), floor(s / len(groups))) W += random_sample(group, t) if len(W) < s: W += random_sample(C - W, s - len(W)) selected = previous_ranks + high_crowding_values + W

(Doerr et al., 16 Dec 2024)

3. Proven Runtime Guarantees for Balanced NSGA-II

The "balanced" tie-breaking enables polynomial-time convergence on benchmarks where standard NSGA-II exhibits exponential slowdown. Let $M$ be the size of the Pareto front, $S$ the maximum incomparability set, and $n' = n/(m/2)$ ($n$ = problem dimensionality; $m$ = objectives):

Many-Objective (m ≥ 3) Polynomial Bounds

For OneMinMax ($m$-objective): Expected generations $\leq 2 e n M$.

For LeadingOnesTrailingZeros ($m$-objective): $\leq 2 e n M + 2 e n^2$.

For OneJumpZeroJump$_k$ ($m$-objective): $\leq 2 e n^k M + 2 e k (m/2) n$.

The required population is $N \geq S + 4n + 2m$, which is polynomial in $n$ for constant $m$ (Doerr et al., 16 Dec 2024).

Bi-Objective Improved Bounds

• OneMinMax: Runtime $O(N n + n^2 \log n)$. For minimal $N = \Theta(n)$, recovers $O(n^2 \log n)$.
• OneJumpZeroJump$_k$: Runtime $O(\max\{n^{k+1}, N n\})$. For $N = \Theta(n)$, recovers $O(n^{k+1})$ vs. classical $O(N n^k)$.
• LeadingOnesTrailingZeros: $O(n^3 + N n \log(N/n))$.

Significantly, runtime plateaus for $N$ in a wide range (e.g., $N = \Theta(n)$ to $N = \Theta(n^k)$ in OneJumpZeroJump$_k$), eliminating the linear-in-$N$ penalty of the classic version (Doerr et al., 16 Dec 2024).

4. Empirical and Theoretical Impact

Empirical results confirm that with balanced NSGA-II, increasing $N$ moderately above the cardinality of the Pareto front yields only a mild increase in computational cost (function evaluations), whereas the classic NSGA-II can suffer drastic slowdowns (Doerr et al., 16 Dec 2024).

The following table summarizes asymptotic runtimes:

Benchmark	Standard NSGA-II	Balanced NSGA-II
OneMinMax (bi)	$\Theta(N n \log n)$	$O(N n + n^2 \log n)$
LOTZ (bi)	$\Theta(N n^2)$	$O(n^3 + N n \log(N/n))$
OJZJ$_k$ (bi)	$\Theta(N n^k)$	$O(N n + n^{k+1})$
OMM ($m\geq 3$)	$\exp(\Omega(n^{\lceil m/2\rceil}))$	$O(n M)$
LOTZ ($m\geq 3$)	suspected exp	$O(n M + n^2)$
OJZJ$_k$ ($m\geq 3$)	unknown/exp	$O(n^k M + n)$

Adding the "prefer rare objective values" rule fixes both the exponential runtime in many-objective settings and the inefficiency for large $N$ in the bi-objective regime (Doerr et al., 16 Dec 2024).

5. Analysis Techniques and Theoretical Insights

All proofs in (Doerr et al., 16 Dec 2024) crucially exploit the following properties:

• Persistence Lemma: Once a given objective vector enters the first rank/front, its frequency in the population does not drop below a positive bound, provided survivor selection preferences rare values rather than breaking ties at random.
• Drift and Coupon-Collector Arguments: Population diversity, maintained by the rarity-based tie-breaker, ensures that discovering all members of the Pareto front proceeds geometrically (i.e., in expected $O(M)$ rounds) rather than being repeatedly set back by random culling.
• Breakdown of Classic NSGA-II: On discrete fronts with large numbers of ties, classic NSGA-II's random tie-breaking loses rare values, leading to plateaus or exponential expected times to reach full coverage. Balanced NSGA-II blocks this.

6. Contextualization and Relation to Broader MOEA Landscape

These results address and mathematically resolve two root deficiencies that had impeded the scalability of NSGA-II in discrete high-dimensional objective settings. The rarity-based tie-breaking yields provable polynomial runtime even where hypervolume- or reference-point-based methods (e.g., SMS-EMOA, NSGA-III) had previously been required for tractable scaling. The approach achieves this with a minimal code change and negligible computational overhead, making it a drop-in enhancement for existing NSGA-II implementations (Doerr et al., 16 Dec 2024).

The theoretical guarantees align with recent advances in runtime analysis of MOEAs and clarify the connection between survivor selection micro-mechanisms and macroscopic scalability. Unlike classic crowding distance, rarity-based selection directly maintains objective space coverage, providing deterministic guarantees for both diversity and convergence in a wide range of multi-objective combinatorial settings.