CEC2013MMO

Problem Difficulty Classification

Difficulty Mode

Training Set

Testing Set

easy

8, 9, 13-20

1-7, 10-12

difficult

1-7, 10-12

8, 9, 13-20

Note: When difficulty is ‘all’, both training and testing sets contain all problems (1-20).

CEC2013MMO is based on CEC2013LSGO benchmark and specially crafeted for multi-modal optimization, which includes 20 synthetic problem instances covering various dimensions (1D~20D), each with varied number of (1 ~ 216) global optima. Among them, F1 to F5 are simple uni-modal functions, F6 to F10 are dimension-scalable functions with multiple global optima, and F11 to F20 are complex composition functions with challenging landscapes.

Characteristics

  • Original tasks are maximization; converted to minimization via sign flip.

  • Covers 1D to 20D settings with varying numbers of optima.

Function Summary

ID Function Name Dim Global Optima Local Optima Range
F1 Five-Uneven-Peak Trap 1 2 3 [0, 30]
F2 Equal Maxima 1 5 0 [0, 1]
F3 Uneven Decreasing Maxima 1 1 4 [0, 1]
F4 Himmelblau 2 4 0 [-6, 6]
F5 Six-Hump Camel Back 2 2 2 [-1.1, 1.1]
F6 Shubert D 3^D many [-10, 10]^D
F7 Vincent D 6^D 0 [0.25, 10]^D
F8 Modified Rastrigin D $$\prod_{i = 1}^{D} k_i$$ 0 [0, 1]^D
F9–F12 Composition Functions 2–20 6–8 complex [-5, 5]^D

Dataset Setup

The parameter settings used for each problem are as follows. Note again that here we reformulate the origin problems as minimization problems in the dataset setting by applying negative signs to the evaluation results of the original functions.

Problem id Function r Peaek height No. global optima
P1 F1 (1D) 0.01 -200.0 2
P2 F2 (1D) 0.01 -1.0 5
P3 F3 (1D) 0.01 -1.0 1
P4 F4 (2D) 0.01 -200.0 4
P5 F5 (2D) 0.5 -1.031628453 2
P6 F6 (2D) 0.5 -186.7309088 18
P7 F7 (2D) 0.2 -1.0 36
P8 F6 (3D) 0.5 -2709.093505 81
P9 F7 (3D) 0.2 -1.0 216
P10 F8 (2D) 0.01 2.0 12
P11 F9 (2D) 0.01 0 6
P12 F10 (2D) 0.01 0 8
P13 F11 (2D) 0.01 0 6
P14 F11 (3D) 0.01 0 6
P15 F12 (3D) 0.01 0 8
P16 F11 (5D) 0.01 0 6
P17 F12 (5D) 0.01 0 8
P18 F11 (10D) 0.01 0 6
P19 F12 (10D) 0.01 0 8
P20 F12 (20D) 0.01 0 8

Evaluation Metrics

  • Peak Ratio (PR): Measures average % of known optima found.

  • Success Rate (SR): % of runs that find all global optima.

  • Both metrics use ε = 1e-4 as the primary accuracy threshold.

Max Function Evaluations (MaxFEs)

Function Range MaxFEs
F1–F5 (1D/2D) 5e4
F6–F11 (2D) 2e5
F6–F12 (≥3D) 4e5

Train-Test Split

Based on difficulty from empirical studies:

  • Easy Problems: P1–P7, P10–P12

  • Difficult Problems: P8–P9, P13–P20

Mode Train Set Test Set
Easy Difficult Set Easy Set
Difficult Easy Set Difficult Set