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This paper analyzes the Strength Pareto Evolutionary Algorithm 2 (SPEA2), a widely used multi-objective optimizer. The core idea is to mathematically study SPEA2's complex selection mechanism, particularly its balanced selection property, which differs from NSGA-II. The main empirical claim is that SPEA2 achieves improved runtime guarantees for benchmark problems like ONEMINMAX, LEADINGONESTRAILINGZEROS, and ONEJUMPZEROJUMP, with runtimes that are less sensitive to the choice of population size compared to previous bounds for SPEA2 and NSGA-II.

The paper addresses several bottlenecks and gaps in the theoretical understanding of multi-objective evolutionary algorithms (MOEAs):

Limited theoretical understanding of SPEA2: SPEA2 is one of the most widely used domination-based MOEAs, but its complex selection mechanism makes it one of the least understood from a theoretical perspective.
Linear dependence of runtime on population size: Previous runtime guarantees for SPEA2 and NSGA-II show a linear dependence on the population size, meaning increased population size weakens performance guarantees.

Proof of balanced selection: The paper proves that SPEA2's selection mechanism, unlike NSGA-II, removes individuals with the most present objective value, leading to a more balanced distribution of objective values on the Pareto front.
Improved runtime guarantees: New runtime guarantees are proven for SPEA2 on ONEMINMAX, LEADINGONESTRAILINGZEROS, and ONEJUMPZEROJUMP benchmarks, showing less dependence on population size.
Demonstration of robustness to parameter changes: The results suggest that SPEA2 is more robust to changes in its parameters (parent population size μ and offspring population size λ) than other domination-based MOEAs.
Multiplicative growth analysis: The paper shows that the number of individuals with a given objective value exhibits a multiplicative growth behavior, which is key to the improved bounds.

The paper mathematically analyzes the SPEA2 algorithm, which maximizes an m-objective function f: Ω → R^m.

Algorithm Flow:

Initialization: A parent population P₀ of μ individuals is sampled independently and uniformly at random from Ω.
Iteration (t ∈ Z>₀):
- Offspring Generation: An offspring population Q_t is created by selecting λ parents uniformly at random from P_t and applying standard bit mutation to each.
- Combined Population: The parent and offspring populations are combined into R_t = P_t ∪ Q_t.
- Non-dominated Selection: Non-dominated individuals S_t are identified from R_t.
- Population Reduction (if |S_t| > μ): If the number of non-dominated individuals exceeds the parent population size μ, individuals are removed from S_t one by one using the eliminate function based on the σ-criterion until |S_t| = μ.
- Population Expansion (if |S_t| < μ): If the number of non-dominated individuals is less than μ, dominated individuals from R_t \ S_t are added to S_t using the add_dominated function until |S_t| = μ.
- Next Parent Population: P_{t+1} is set to S_t.

σ-criterion: This criterion is used by the eliminate function to remove individuals. For a multi-set A of elements from Ω, and an individual x ∈ A, σ_k(x) is defined as the distance between f(x) and its k-th nearest neighbor in f(A). Individuals are sorted lexicographically based on {(σ_k(x))_k∈[1..|A|−1]}. The eliminate function removes an element x from A that is minimal with respect to this lexicographic order, breaking ties uniformly at random. This criterion favors individuals further away from their neighbors in the objective space.

Multiplicative Growth: The analysis builds on the balanced selection property of SPEA2 to show that the number of individuals with a given objective value exhibits a multiplicative growth behavior, using a bound on the expected hitting time of a stochastic process with multiplicative growth.

Population Sizes:
- Parent population size: μ ∈ Z>1
- Offspring population size: λ ∈ Z>1
Search Space: Ω = {0,1}^n (bit strings of length n ∈ Z>1)
Mutation: Standard bit mutation, where each bit of a selected individual is flipped independently with probability 1/n.
σ-criterion: For a multi-set A and x ∈ A, σ_k(x) is the distance between f(x) and its k-th nearest neighbor in f(A). The relation ≤_d is the lexicographic order over {(σ_k(x))_k∈[1..|A|−1]}. An element x is removed if it is minimal with respect to ≤_d.
Key Equations (Lemma 2 for multiplicative growth):
- X₀ = 1
- Y_{t+1} Bin(k, min{rX_t, 1})
- X_{t+1} = X_t + Y_{t+1}
- Expected hitting time E[T] ≤ 4[log_{1+kr} B] for X_t to reach B.
Benchmarks:
- ONEMINMAX (OMM): f: {0,1}^n → R², x → (|x|₀, |x|₁). Pareto front: {(k, n − k) | k ∈ [0..n]}.
- ONEJUMPZEROJUMP (OJZJ_k): f: {0,1}^n → R², x → (f₁(x), f₀(x)), where f_i(x) = { k + |x|_i if |x|_i ≤ n - k or |x|_i = n, n - |x|_i otherwise. Gap size k ∈ [2..[n/2]]. Pareto front: F* = {(i, n + 2k - i) | i ∈ [2k..n] ∪ {k,n + k}}.
- LEADINGONESTRAILINGZEROS (LOTZ): f: {0,1}^n → R², x → (f₁(x), f₂(x)), where f₁(x) = Σ_{i=1}^n Π_{j=1}^i x_j and f₂(x) = Σ_{i=1}^n Π_{j=0}^{i-1} (1 - x_{n-j}). Pareto front: {(k, n − k) | k ∈ [0..n]}.

ONEMINMAX (OMM):
- Conditions: Parent population size μ > n + 1, offspring population size λ.
- Runtime: Expected O((μ + λ)n + n² log n) function evaluations.
- Comparison: This is an asymptotic runtime gain compared to the previous best bound of O(μη log n) (for λ = O(μ) only). For n + 1 < μ = O(n log n) and λ = O(n log n), it yields O(n² log n), which is strictly better than NSGA-II's O(μη log n) when μ = ω(n).
LEADINGONESTRAILINGZEROS (LOTZ):
- Conditions: Parent population size μ > n + 1, offspring population size λ.
- Runtime: Expected O((μ + λ)n log n + n³ + λn) function evaluations.
- Comparison: This is O(n³) for n + 1 ≤ μ = O(n²) and λ = O(n²), which is the typical runtime of standard MOEAs on this benchmark.
ONEJUMPZEROJUMP (OJZJ_k):
- Conditions: Gap size k ∈ [2, [n/2]] ∩ Z, parent population size μ > n - 2k + 3, offspring population size λ.
- Runtime: Expected O((μ + λ)n + n^{k+1}) function evaluations.
- Comparison: The typical runtime of O(n^{k+1}) is achieved for the large range of all n - 2k + 3 < μ = O(n^k) and all λ = O(n^k).

Computational Cost of σ-criterion: The σ-criterion, while theoretically beneficial for balanced selection, is computationally more costly than the crowding distance used in NSGA-II.
Applicability to small population sizes: The reported results do not apply when the population size is smaller than the Pareto front size.
Theoretical nature: The work is purely theoretical, and there is no information on how modifications (e.g., to NSGA-II) influence practical performance.

Method evidence: "The function eliminate removes one individual from the given multi-set, according to a concept called σ-criterion. The σ-criterion favors individuals that are further away from their neighbors, in the objective space." (section: The SPEA2)
Result evidence: "For ONEJUMPZEROJUMP problems with gap size k ∈ [2,2] ∩ Z, the improvement is even more drastic. We prove an expected runtime of O((μ + λ)n + nk+1) function evaluations (Theorem 7), for μ > n − 2k + 3." (section: Abstract)
Limitation evidence: "We note that the selection in the SPEA2 based on the σ-criterion is computationally more costly than the crowding distance employed by the NSGA-II." (section: Conclusion)

Improved Runtime Guarantees for the SPEA2 Multi-Objective Optimizer

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