Random Optimization Sequences

Updated 15 November 2025

Random Optimization Sequences are defined as vectors or scalars generated via PRNGs, chaotic maps, or LDS to inject controlled stochasticity in optimization algorithms.
Their statistical properties, particularly the amplitude distribution, directly affect convergence rates and solution quality in methods like PSO.
Integrating LDS in PSO variants enhances search uniformity and reduces iteration counts by up to 50%, ensuring efficient and robust convergence.

Random optimization sequences play a pivotal role in metaheuristic algorithms, especially those designed for continuous optimization such as Particle Swarm Optimization (PSO). These sequences, which can be generated via pseudo-random number generators (PRNG), chaotic maps, or deterministic low-discrepancy sequences (LDS), determine the stochastic components that drive exploration and exploitation dynamics. Their statistical and dynamical properties directly impact algorithmic convergence, search coverage, and solution robustness.

1. Random and Chaotic Sequences: Definitions and Mathematical Properties

A random optimization sequence refers to a vector-valued or scalar-valued sequence $\{r_n\}$ used to inject stochasticity into the update rules of optimization algorithms. In PSO and its variants, such sequences are most commonly sampled from uniform or other parametric distributions over $[0,1]$ . Chaotic sequences, in contrast, are generated by iterating deterministic nonlinear maps exhibiting sensitive dependence on initial conditions but producing stationary invariant densities. Two prominent classes considered in algorithmic optimization studies are:

PRNG-based sequences: Instances include independent draws from $\mathcal{U}[0,1]$ , Beta, or truncated Normal distributions.
Chaotic map sequences: These involve recursive updates such as the Logistic map $z_{n+1}=r\,z_n(1-z_n)$ ( $r=4$ ), Cubic map $z_{n+1}=r\,z_n(1-z_n^2)$ ( $r=2.62$ ), or Bellows map $z_{n+1}=r\,z_n(1+z_n)^{-6}$ ( $r=2.0$ ), each generating a unique invariant density over $(0,1)$ .

The mathematical distinction concerns not only the origin of the sequence but also the density function $\varrho(z)$ describing the long-run statistical distribution of generated points, which shapes algorithmic behavior.

2. Decoupling Sequence Origin from Statistical Distribution

Recent empirical work rigorously differentiates the effects attributable to the origin of a sequence—random versus chaotic—from those attributable to its statistical distribution, particularly in the context of PSO (Nörenberg et al., 2023). By selecting pairs of chaotic maps and random distributions that induce identical density functions (e.g., the Logistic map and the Beta $(0.5,0.5)$ distribution), it is possible to isolate and compare performances with only the sequence origin variable changed. Conversely, keeping the origin fixed and altering the underlying density (e.g., $\mathcal{B}(0.5,0.5)$ vs. $\mathcal{B}(1,5)$ among random sequences) elucidates the impact of distribution shape on search outcomes.

Key finding: whenever two sequences share the same stationary density—even if one is chaotic and one is random—algorithmic performance is statistically indistinguishable. By contrast, differences in the probability distribution yield observable performance changes, regardless of sequence origin. This suggests that the amplitude distribution's shape, rather than dynamical chaos, governs convergence rates and solution quality.

3. Low-Discrepancy Sequences and Their Effects in Optimization

Low-discrepancy sequences (LDS), such as Sobol', Halton, optimized Halton (OHS), and dynamic-evolution sequences (DES), are deterministic yet designed to uniformly fill the $[0,1]^D$ hypercube with quantifiably minimal clustering or gaps. Replacement of random multipliers in velocity update equations with LDS terms (e.g., $\mathbf{r}_k^g \leftarrow \mathbf{u}^{\,\alpha_k(g)}$ where $\mathbf{u}^n$ is the $n$ th point in the LDS) preserves uniform coverage of stochastic search directions throughout the optimization process (Zhao et al., 2022).

Unlike PRNGs, for which $N$ -point discrepancy scales as $O(N^{-1/2})$ , classic LDS offer bounds of $D_N^*=O((\log N)^D/N)$ , ensuring much fuller coverage at practical sample sizes. This property addresses the loss of coverage over iterations due to stochastic "clumping," stabilizing exploration and reducing superfluous iterations.

4. Practical Implications and Empirical Findings in PSO and HCLPSO

Empirical evaluation of both standard PSO and Heterogeneous Comprehensive Learning PSO (HCLPSO) variants reveals the implications of sequence properties for optimization performance:

Random vs. Chaotic Sequences in PSO

PSO update formulas leverage $r_1,r_2$ $r_{1}, r_{2}$ coefficients sampled from either PRNG-based or chaotic sources. Systematic benchmarking (15 IEEE-CEC 2013 functions, 1000 runs per setting) using statistical tests (Wilcoxon rank-sum) demonstrates that:
- Performance is virtually identical ([2,1,12]: better, worse, indistinguishable out of 15 functions) for Logistic vs. Beta $(0.5,0.5)$ , which share the density but differ in sequence origin.
- Larger differences appear only when the density changes, e.g., Logistic vs. Beta $(1,5)$ ([5,6,4] count), regardless of sequence origin.
Conclusion: The distribution of $r_1, r_2$ dominates, and the sequence origin (chaotic vs. random) is secondary. There is no intrinsic benefit to chaos unless the invariant density is better matched to the optimization landscape.

Low-Discrepancy Sequences in HCLPSO

Replacing random velocity multipliers ( $\mathbf{r}_1^g, \mathbf{r}_2^g$ $r_{1}^{g}, r_{2}^{g}$ ) with LDS (DES, OA, OHS, HWS) yields:
- Statistically significant reductions in required iterations for convergence (avg. rank improvements: e.g., OA 1.00 at $\varepsilon=5\%$ vs. random 4.88).
- No compromise in success rates across $D=10, 30, 50$ dimensional benchmarks; LDS variants maintain or exceed performance on all tested CEC'17 functions.
- On composite, difficult test cases, LDS-based HCLPSO often cuts the number of iterations by 30–50%, with steepened convergence profiles.

A plausible implication is that maintaining the uniformity of stochastic directions throughout the entire optimization run—not just at initialization—directly translates to efficient search and robust convergence.

5. Statistical Evaluation and Algorithmic Recommendations

Empirical results across papers consistently employ rigorous statistical testing (e.g., the Friedman and Nemenyi tests at 95% confidence) and focus on metrics including mean error to optimum, number of iterations to solution, and success rate under fixed accuracy tolerances.

Summary table of notable findings:

Setting	Performance Outcome	Origin Effect	Distribution Effect
Logistics vs. Beta(0.5,0.5)	No significant difference	None	None
LDS vs. PRNG in HCLPSO	Fewer iterations, matched success rate	None	Uniformity of LDS
Different Beta/N or chaotic	Performance shift as distribution varies	N/A	Marked

Practitioners are advised to prioritize the match between the density of stochastic coefficients and the needs of the optimization problem. For PSO, careful selection or shaping of the multiplier distributions (beta, uniform, truncated normal) should precede considerations of chaos or determinism in the source. In HCLPSO and similar frameworks, LDS-based deterministic deterministic “jitter” is effective for faster, more robust search.

6. Common Misconceptions and Research Trends

Previous literature often reported enhanced PSO performance using chaotic sequences, inferring superiority from empirical gains. The present data show that such effects trace entirely to differences in density functions of the chaotic maps relative to the uniform random distributions typically used, not to any inherent advantage of chaos itself (Nörenberg et al., 2023). This reframes the origin-centric view: as long as the density is reproduced faithfully (chaotic or random or deterministic LDS), optimization results will be comparable.

A continuing research question concerns how best to design, select, or adapt densities and LDS types to problem-specific landscapes, given the demonstrated importance of amplitude distribution and uniformity.

7. Directions for Future Study

Current results point toward two key future directions:

Density Optimization: Systematic procedures for adaptively selecting or evolving the distribution of multipliers ( $r_1$ , $r_2$ ) in response to observed search behavior are not yet standardized.
LDS Hybridization and High-Dimensional Scaling: While LDS superiority in uniformity is clear for moderate $D$ , scaling properties, and the potential benefits of hybrid stochastic-deterministic schemes, merit deeper investigation, especially for irregular, non-convex, or dynamically changing landscapes.

A plausible implication is that further gains in metaheuristics may result, not from novel sequence origins, but from active control over the statistical properties of injected noise and from sequence designs that harness the superior coverage properties of LDS.