A Longitudinal Analysis of the CEC Single-Objective Competitions (2010-2024) and Implications for Variational Quantum Optimization

Vojtěch Novák; Tomáš Bezděk; Ivan Zelinka; Swagatam Das; Martin Beseda

quant-phMar 25, 2026

A Longitudinal Analysis of the CEC Single-Objective Competitions (2010-2024) and Implications for Variational Quantum Optimization

Authors: Vojtěch Novák, Tomáš Bezděk, Ivan Zelinka, Swagatam Das, Martin Beseda

Organizations: Department of Computer Science, Faculty of Electrical Engineering and Computer Science, VSB - Technical University of Ostrava, Ostrava, Czech Republic · IT4Innovations National Supercomputing Center, VSB - Technical University of Ostrava, 708 00 Ostrava, Czech Republic · Department of Informatics and Statistics, Marine Research Institute, Klaipeda University, Lithuania · Department of Mathematics, TUM School of Computation, Information and Technology, Technical University of Munich, Garching bei München, Germany ·

Abstract

This paper provides a historical analysis of the IEEE CEC Single Objective Optimization competition results (2010-2024). We analyze how benchmark functions shaped winning algorithms, identifying the 2014 introduction of dense rotation matrices as a key performance filter. This design choice introduced parameter non-separability, reduced effectiveness of coordinate-dependent methods (PSO, GA), and established the dominance of Differential Evolution variants capable of preserving the rotational invariance of their difference vectors, specifically L-SHADE. Post-2020 analysis reveals a shift towards high complexity hybrid optimizers that combine different mechanisms (e.g., Eigenvector Crossover, Societal Sharing, Reinforcement Learning) to maximize ranking stability. We conclude by identifying structural similarities between these modern benchmarks and Variational Quantum Algorithm landscapes, suggesting that evolved CEC solvers possess the specific adaptive capabilities required for quantum control.

Figures & tables

Figure 1: Longitudinal evolution of algorithm families in CEC Competitions (2010–2024). Stacked bars illustrate the percentage distribution of all participating entries by category, with text annotations identifying the top-five-ranked algorithms for each year. To address the structural hybridization characteristic of post-2020 solvers, algorithmic classification follows a strict "Primary Search Engine" protocol. Solvers are categorized based on the specific evolutionary mechanism that consumes the majority (

>50\%

) of the maximum function evaluation budget. To accurately capture evolutionary trajectories, the DE family is subdivided into Canonical/Adaptive, SHADE Lineage, Societal/Multi-Population, and Driven Ensembles. Under this protocol, multi-component architectures are classified by their dominant foundational logic; for instance, ensembles that rely on DE as their core exploration framework but periodically trigger Covariance Matrix Adaptation to navigate rotated valleys (e.g., EA4eigN100 ) are classified as DE-Driven Ensembles rather than generic hybrids. This granular taxonomy confirms a structural shift from early algorithmic diversity to the sustained dominance of highly specialized DE architectures.

Rank	CEC 2010 (LSGO)	CEC 2011	CEC 2013 (N M-O)	CEC 2013 (R-P SO)
1	MA-SW-Chains [ 132 ]	GA-MPC [ 59 ]	NEA2 [ 144 ]	ICMAES-ILS [ 115 ]
2	2-Stage Ensemble [ 203 ]	SAMODE [ 58 ]	dADE/nrand/1 [ 61 ]	NBIPOPaCMA [ 118 ]
3	MultiPSO+Harmony [ 105 ]	EA-DE-MA [ 166 ]	CMA-ES	DRMA [ 102 ]
4	DASA [ 97 ]	WI-DE [ 78 ]	N-VMO [ 17 ]	SHADE [ 183 ]
5	SADE [ 33 ]	Adap. DE 171 [ 8 ]	dADE/nrand/2 [ 61 ]	LS-MPC [ 60 ]
6	DECC-D [ 138 ]	DE- $\Lambda$ [ 149 ]	DE/nrand/2 [ 62 ]	–

Table 1: CEC 2010 Large Scale Global Optimization Challenge, CEC 2011 Competition on Testing Evolutionary Algorithms on Real World Optimization Problems and CEC 2013 – Rankings (N MO means Niching Methods for Multimodal Optimization, R-P means Real-Parameter Single Objective)

Rank	CEC 2014	CEC 2015	CEC 2016
1	L-SHADE [ 182 ]	SPS-L-SHADE-EIG [ 74 ]	LSHADE-EpSin [ 11 ]
2	GaAPADE [ 208 ]	DEsPA [ 185 ]	UMOEA-II [ 57 ]
3	MVMO-SH [ 64 ]	LSHADE-ND [ 142 ]	iL-SHADE [ 34 ]
4	OptBees [ 121 ]	L-SHADE [ 182 ]	LSHADE44 [ 143 ]
5	RM-MEDA [ 217 ]	MVMO-PH [ 151 ]	L-SHADE [ 182 ]
6	UMOEAs [ 56 ]	DE_b6e6rl [ 196 ]	ECL-L-SHADE [ 110 ]

Table 2: CEC 2014, CEC 2015 and CEC 2016 – Main Track Rankings

Rank	CEC 2017	CEC 2018	2019 (100-D)
1	EBOwithCMAR [ 99 ]	HS-ES [ 214 ]	jDE100 [ 36 ]
2	jSO [ 35 ]	LSHADE-ESP [ 169 ]	DISHchain1e+12 [ 213 ]
3	LSHADE-SPACMA [ 76 ]	ELSHADE_SPACMA [ 77 ]	HyDE-DF [ 109 ]
4	LSHADE-cnEpSin [ 14 ]	EBOwithCMAR [ 99 ]	SOMA T3A [ 52 ]
5	DES [ 13 ]	UMOEAsII [ 158 ]	LSHADE-USM [ 130 ]
6	MM-OEDA [ 13 ]	MVMO-PH [ 152 ]	SOMA Pareto [ 190 ]

Table 3: Algorithm Rankings Part 1: CEC 2017 – 2019

Rank	CEC 2020	CEC 2021	CEC 2022	CEC 2024
1	IMODE [ 159 ]	NL-SHADE-RSP [ 172 ]	EA4eigN100 [ 37 ]	L-SRTDE [ 173 ]
2	AGSK [ 128 ]	jDE21 [ 5 ]	NL-SHADE-LBC [ 168 ]	RDE [ 189 ]
3	j2020 [ 3 ]	APGSK_IMODE [ 70 ]	NL-SHADE-RSP-MID [ 18 ]	BlockEA [ 147 ]
4	OLSHADE [ 171 ]	DEDMNA [ 98 ]	S_LSHADE_DP [ 198 ]	mLSHADE [ 45 ]
5	jDE100e [ 4 ]	MadDE [ 27 ]	jSObinexpEig [ 95 ]	jSO [ 40 ]
6	RASP-SHADE [ 156 ]	MLS-LSHADE [ 26 ]	MTT_SHADE [ 179 ]	iEACOP [ 188 ]

Table 4: Algorithm Rankings Part 2: CEC 2020 – CEC 2024

Function	HS-ES (ES/CMA)	LSHADE- RSP (DE)	ELSHADE- SPACMA (DE)	UMOEAsII (Hybrid)	MVMO- PH (Hybrid)
Unimodal ( $F_{1},F_{3}$ )	4.00	1.00	1.00	5.00	1.00
Multimodal ( $F_{4}-F_{10}$ )	1.57	3.14	3.14	2.29	4.71
Hybrid ( $F_{11}-F_{20}$ )	1.80	2.10	2.60	3.90	4.50
Comp. ( $F_{21}-F_{30}$ )	2.90	2.50	2.70	3.00	3.90
Overall Rank	2.29	2.36	2.57	3.39	4.07

Table 5: Average Rank per Function Class (CEC 2018, 50-Dimensions). Data demonstrates the performance shift across landscape complexities, justifying the evolutionary drive toward hybridization on Composition functions.

Function Group	jDE100 (DE-R1)	DISHchain 1e+12 (DE-R2)	HyDE-DF (DE-R3)	SOMA T3A (Swarm-R3)	ESHADE- USM (DE-R5)
Solved ( $F_{1}\text{--}F_{7},F_{10}$ )	10.00	10.00	10.00	10.00	10.00
Extr. Persist. ( $F_{8}$ )	10.00	10.00	10.00	10.00	2.00
Cond. Trench ( $F_{9}$ )	10.00	7.12	3.00	3.00	3.52
Total (Max 100)	100.00	97.12	93.00	93.00	85.52

Table 6: Average Correct Digits Found (CEC 2019, 100-Digit Challenge). The table highlights how the top algorithms easily solved

F_{1}-F_{7},F_{10}

, leaving

F_{8}

(Expanded Schaffer) and

F_{9}

(Happy Cat) as the ultimate architectural filters. SOMA T3A emerges as the only non-DE algorithm in the top tier.

Algorithm	Family	Non-Shifted Rank (Opt. at Origin)	Rotated/Shifted Rank (True Invariance)
APGSK-IMODE	DE (Multi-Pop)	1	4
MadDE	DE (Adaptive)	2	6
LSHADE	DE (SHADE)	3	8
NL-SHADE-RSP	DE (SHADE)	5	1
jDE21	DE (Adaptive)	7	2
DEDMNA	DE (Adaptive)	6	3

Table 7: Rank Inversion in CEC 2021 Parameterized Benchmarking. The table illustrates the “Center-Bias” effect. Center-biased hybrids (APGSK-IMODE) dominate when the optimum is at the origin (Non-Shifted). However, when the optimum is shifted and rotated, true rotationally invariant DE lineages (NL-SHADE-RSP) reclaim dominance, while pure CMA-ES (RB-IPOP-CMA-ES) fails consistently.

Function	L-SRDE (DE-R1)	RDE (DE-R2)	mLSHADE (DE-R4)	EACOP (Hybrid-R6)
$f_{21}$	3	4	2	6
$f_{22}$	3	2	5	1
$f_{23}$	3	4	1	2
$f_{24}$	3	1	2	6
$f_{25}$	3	6	5	1
$f_{26}$	3	1	2	6

Table 8: U-Score Rankings on CEC 2024 Composition Functions. The U-score evaluates both convergence speed and final accuracy. The data demonstrates that the winning algorithm, L-SRDE, achieved victory through extreme consistency across non-separable topologies, whereas the hybrid EACOP exhibited severe polarization (alternating between 1st and 6th place).

Domain	Methods	Noise Sources	Key Benchmarks
Quantum Chemistry	VQE, ADAPT-VQE, SAOOVQE	Gate infidelity, decoherence, shot noise	Chem. accuracy (1.6 mHa) on SC [ 165 , 186 ]
Optimization	QAOA, RED-QAOA, Recursive	Circuit depth, spectral gaps	Advantage over random; warm-starting [ 161 ]
Machine Learning	QCNN, VQC, Hybrid Transfer	Barren plateaus, sampling noise	High effective dim. vs. classical NN [ 41 ]
Finance	VQE for QUBO, Portfolio Opt.	Landscape roughness, volatility	Arbitrage/portfolio on NISQ [ 54 ]
Materials	Ham. Simulation, Periodic VQE	Many-body complexity, discretization	Ground state for correlated models [ 42 ]
Early Fault Tolerance	EFT-VQA, PEC, pQEC	Physical error rate ( $\sim 0.1\%$ ), QEC overhead	9.27x fidelity gain via pQEC [ 48 , 114 ]

Table 9: VQA Optimization Under Quantum Noise

Figure 2: Publication volume across three primary search clusters. The blue line establishes the baseline of the optimization community. The orange line tracks the “Quantum Revolution”, starting around 2015-2016 when NISQ devices became a major research focus. The green line illustrates the intersection of quantum computing and optimization.

Figure 3: Parameter concentration in QAOA for Max-Cut on 3-regular graphs with unweighted ( $w_{ij}=1$ ) and weighted ( $w_{ij}\in\{\pm 1,\pm 2,\pm 3\}$ ) edges using circuit depth $p=4$ . (a) PCA visualization of optimization trajectories identifying local minimum (blue), maximum (red), and random initialization (white) points on a 10-qubit graph instance. Panel (b) reveals that the local minima (left columns) possess non-separable, rotated valley structures that persist across increasing problem dimensions ( $N=4$ to $20$ ). Source: [ 150 ]

CEC Feature	Quantum Analog	Physical Origin	Algorithmic Need
Multimodal	Rugged landscapes, local traps	Frustrated interactions (Hubbard, Spin Glass)	Global search (DE) to escape local basins
Non-Sep.	Parameter Correlation (Entanglement)	Non-commuting gates; $\theta$ dependencies	Rotation-invariant opt. (CMA-ES, DE)
Ill-Cond.	Barren Plateaus, Narrow Gorges	Deep ansatz; Lie algebra redundancies	Hessian/Covariance-based navigation
Noisy	Stochastic Shot Noise	Finite sampling	Noise-tolerant search

Table 10: CEC Benchmark Features vs. Quantum landscapes

Panel of Figure 4 — Figure 4: Hardware-efficient ansatz with $L=1$ entangling layer for $N=10$ qubits. The circuit utilizes $R_{y}$ rotation gates and a linear CNOT staircase to produce a 20-dimensional parameter space.

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Evolutionary optimization searches for strong solutions through population-based variation, selection, and adaptation.

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Quantum Approximate Optimization Algorithm Variational Quantum Algorithms Loss Landscape Geometry

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Variational quantum algorithms turn choices of Hamiltonian, ansatz, and parameterization into a classical nonconvex optimization problem. We study how this objective function can be visualized and characterized in ways that help explain optimizer behavior. We distinguish two properties of the objective: the number of local minima encountered along sampled directions and the differences in quality among local-search endpoints. We then ask how a local optimizer, represented by BFGS, compares with adaptive differential evolution, represented by jSO. Rather than comparing jSO with a single local run, we allow BFGS multiple starts within the same function-evaluation budget. This gives local search repeated opportunities to explore different basins and provides a stronger baseline for asking when a global evolutionary solver is useful. We study these questions using VQE and QAOA, focusing on how frustra- tion, circuit depth, parameter tying, mixed locality, and nonlinear repa- rameterization change the Hamiltonian expectation-value objective seen by the classical optimizer. Increasing independent circuit depth raises the sampled local-minimum count without making global search more effective. Parameter tying, by contrast, produces both more repeated local structure and much larger differences in quality among local-search endpoints, and in this regime adaptive differential evolution outperforms function-evaluation-matched multistart BFGS. The comparison shows that the number of local minima alone does not determine whether global search is advantageous: the important distinction is whether different basins lead to similarly good solutions or to substantially different objective values. These results provide a practical workflow for connecting model construction, objective- function geometry, empirical diagnostics, and optimizer choice.

Abstract

Evolutionary Optimization

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