A Novel Approach to Enhance DIRECT-Type Algorithms for Hyper-Rectangle Identification
Abstract
:1. Introduction
- Dedicated Vertex Database Integration: This work introduces a new approach by incorporating a dedicated vertex database. The purpose of this strategic embedding is to constrain sampling points within descent sub-regions, effectively mitigating the risk of oversampling. By implementing this vertex database, the algorithm provides more efficient computations, improving both the accuracy and speed of the optimization process.
- Innovative Grouping Strategy for Hyper-Rectangle Identification: A groundbreaking grouping strategy is introduced, specifically designed for the efficient identification of hyper-rectangles in DIRECT-type algorithms. This innovation addresses the challenge of managing and organizing data points in the search space. By taking advantage of this advanced clustering strategy, the algorithm optimizes the hyper-rectangle identification process, resulting in a more rational and powerful exploration of the solution space.
- Performance Enhancement in the BIRECTv Algorithm: This paper outlines the significant improvements made to the BIRECTv algorithm. These developments have a positive and noticeable impact on the overall efficiency of the algorithm. By refining the BIRECTv approach, this research contributes to a more robust and efficient optimization algorithm, with advances in convergence rates and solution quality. The refined BIRECTv algorithm illustrates the practical implications of the suggested contributions in the design of the underlying optimization process.
2. Overview of Existing Methods for Selecting (POHs) in Various DIRECT-Type Approaches
3. From BIRECT to BIRECTv(impr.)
3.1. The Original BIRECT
3.1.1. Selection Criterion
- At each iteration (kth iteration), starting from the current partition
- The identification of a potentially optimal hyper-rectangle (POH) is based on lower bound estimates of the objective function over each hyper-rectangle, with a fixed rate of change (analogous to a Lipschitz constant).
- A hyper-rectangle , is considered potentially optimal if specific inequalities involving (a positive constant) and the current best-known function value are satisfied.
Algorithm 1 Main steps of the BIRECT algorithm |
|
3.1.2. Division and Sampling Criterion
- After the initial partitioning, BIRECT proceeds to future iterations by partitioning POHs and evaluating the objective function at new sampling points.
- New sampling points are generated by adding and subtracting a distance equal to half the side length of the branching coordinate from the previous points. This approach allows for the reuse of old sampled points in descendant subregions.
- An important aspect of the algorithm is how the selected hyper-rectangles are divided. For each POH, the set of maximum coordinates (edges) is computed, and the hyper-rectangle is bisected along the coordinate (branching variable ) with the largest side length (). Starting from the coordinates associated with the smallest index j (in case multiple coordinates are eligible):
3.2. Description of the BIRECTv Algorithm
3.3. Integration Scheme for Identification of Potentially Optimal Hyper-Rectangles in DIRECT-Based Frameworks
- Tolerance :
- A tolerance of (0.01) means that the algorithm will consider hyper-rectangles whose and values are within 0.01 of each other.
- This allows for a relatively larger difference between and , meaning the algorithm will be more lenient in selecting POHs.
- This might result in a larger set of POHs, including some with relatively larger differences in their norm values.
- Tolerance :
- A tolerance of (0.0000001) means that the algorithm will consider hyper-rectangles whose and values are within of each other.
- This uses a much smaller tolerance, making the algorithm much stricter in selecting POHs.
- This will result in a smaller set of POHs, only including those with extremely close norm values.
Convergence
4. Results and Discussion
4.1. Implementation
Algorithm 2 Find first index within tolerance |
|
4.2. Discussion
- The improved versions of BIRECTv appear to be reliable choices for optimization tasks, as they consistently outperformed the previously published versions and demonstrated competitive performances in terms of both objective value and computational effort.
- The new algorithms, BIRECT-l (new) and BIRECT (new), show promise and are particularly efficient in terms of the number of function evaluations. However, their objective function values may vary depending on the problem.
- The choice of algorithm should be problem-dependent. Some algorithms may be more suitable for specific problem characteristics, such as unimodal or multimodal objective functions, and global or local optimization.
- These sets of information provide a comprehensive assessment of the algorithms’ performance across various aspects, including solution quality and computational efficiency.
4.3. Examining the Success Rate of Algorithms and Function Evaluation Metrics
4.4. Statistical Analysis of the Results
5. Conclusions and Future Prospects
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
Elements of the vector | |
D | Search domain: an n-dimensional hyper-rectangle |
Normalized search space (unit hyper-cube) | |
Hyper-rectangle in normalized search space at iteration k | |
Measure (size) of hyper-rectangle | |
Objective function | |
Known optimal value: | |
Number of function evaluations | |
The best-found function value | |
Hyper-rectangles representing the current partitioning at iteration k | |
L | Lipschitz constant |
Index set identifying the current partition | |
Potentially optimal hyper-rectangle | |
Percent error | |
Tolerance threshold: | |
Current optimal solution vector | |
Current best-known function value: |
Appendix A
Problem No. | Problem Name | Dimension n | Feasible Region | No. of Local Minima | Optimum |
---|---|---|---|---|---|
Ackley | 2, 5, 10 | multimodal | 0.0 | ||
4 | Beale | 2 | multimodal | 0.0 | |
Bohachevsky 1 | 2 | multimodal | 0.0 | ||
Bohachevsky 2 | 2 | multimodal | 0.0 | ||
Bohachevsky 3 | 2 | multimodal | 0.0 | ||
8 | Booth | 2 | unimodal | 0.0 | |
9 | Branin | 2 | 3 | ||
10 | Colville | 4 | multimodal | 0.0 | |
Dixon & Price | 2, 5, 10 | unimodal | 0.0 | ||
14 | Easom | 2 | multimodal | ||
15 | Goldstein & Price | 2 | 4 | 3.0 | |
Griewank | 2 | multimodal | 0.0 | ||
17 | Hartman | 3 | 4 | ||
18 | Hartman | 6 | 4 | ||
19 | Hump | 2 | 6 | ||
Levy | 2, 5, 10 | multimodal | 0.0 | ||
Matyas | 2 | unimodal | 0.0 | ||
24 | Michalewics | 2 | 2! | ||
25 | Michalewics | 5 | 5! | ||
26 | Michalewics | 10 | 10! | ||
27 | Perm | 4 | multimodal | ||
Powell | 4, 8 | multimodal | |||
30 | Power Sum | 4 | multimodal | ||
Rastrigin | 2, 5, 10 | multimodal | |||
Rosenbrock | 2, 5, 10 | unimodal | |||
Schwefel | 2, 5, 10 | unimodal | |||
40 | Shekel, | 4 | 5 | ||
41 | Shekel, | 4 | 7 | ||
42 | Shekel, | 4 | 10 | ||
43 | Shubert | 2 | 760 | ||
Sphere | 2, 5, 10 | multimodal | |||
Sum squares | 2, 5, 10 | unimodal | |||
50 | Trid | 6 | multimodal | ||
51 | Trid | 10 | multimodal | ||
Zakharov | 2, 5, 10 | multimodal |
Problem No. | BIRECT-(New) | BIRECT | DIRECT-l | DIRECT | ||||
---|---|---|---|---|---|---|---|---|
f.eval. | f.eval. | f.eval. | f.eval. | |||||
1 | 202 | 202 | 255 | |||||
2 | 1268 | 1777 | 8845 | |||||
3 | ||||||||
4 | 436 | 436 | 655 | |||||
5 | 468 | 476 | 327 | |||||
6 | 472 | 478 | 233 | 345 | ||||
7 | 480 | 573 | 693 | |||||
8 | 194 | 215 | 295 | |||||
9 | 242 | 242 | 195 | |||||
10 | 794 | 3379 | 6585 | |||||
11 | 722 | 722 | 513 | |||||
12 | ||||||||
13 | ||||||||
14 | 6851 | |||||||
15 | 274 | 274 | 191 | |||||
16 | 5106 | 8379 | 9215 | |||||
17 | 352 | 352 | 199 | |||||
18 | 764 | 764 | 571 | |||||
19 | 196 | 334 | 321 | |||||
20 | 152 | 152 | 105 | |||||
21 | 968 | 1024 | 705 | |||||
22 | 6402 | 7904 | 5589 | |||||
23 | 90 | 94 | 107 | |||||
24 | 126 | 126 | 69 | |||||
25 | ||||||||
26 | ||||||||
27 | ||||||||
28 | ||||||||
29 | ||||||||
30 | ||||||||
31 | 180 | 1727 | 987 | |||||
32 | 1394 | |||||||
33 | ||||||||
34 | 285 | 1621 | ||||||
35 | 1690 | 1700 | 2703 | |||||
36 | ||||||||
37 | 341 | 255 | ||||||
38 | 7210 | |||||||
39 | ||||||||
40 | 1272 | 1200 | 155 | |||||
41 | 1204 | 1180 | 145 | |||||
42 | 1140 | 1140 | 145 | |||||
43 | 2043 | 2967 | ||||||
44 | 118 | 118 | 209 | |||||
45 | 602 | 712 | 4653 | |||||
46 | 8742 | |||||||
47 | 226 | 244 | 107 | |||||
48 | 1000 | 1034 | 833 | |||||
49 | 5538 | 7688 | 8133 | |||||
50 | 1506 | 8731 | 5693 | |||||
51 | ||||||||
52 | 338 | 502 | 237 | |||||
53 | ||||||||
54 | ||||||||
Average | ||||||||
Median |
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Problem No. | BIRECTv-l (impr.) | BIRECTv (impr.) | BIRECTv-l [27] | BIRECTv [27] | BIRECT-l (New) | BIRECT (New) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
f.eval. | f.eval. | f.eval. | f.eval. | f.eval. | f.eval. | |||||||
1 | 153 | 156 | 192 | 134 | 158 | |||||||
2 | 387 | 1135 | 422 | 1578 | 358 | 1062 | ||||||
3 | 1000 | 1000 | ||||||||||
4 | 474 | 742 | 638 | 1034 | ||||||||
5 | 209 | 254 | 284 | 496 | 496 | |||||||
6 | 211 | 252 | 284 | 682 | 682 | |||||||
7 | 209 | 248 | 282 | 852 | 849 | |||||||
8 | 249 | 300 | 334 | 330 | 330 | |||||||
9 | 480 | 370 | 652 | 490 | 242 | |||||||
10 | 1614 | 1337 | 2318 | 1868 | 794 | |||||||
11 | 263 | 431 | 346 | 578 | 234 | |||||||
12 | 2087 | 2652 | 2912 | 6103 | 6125 | |||||||
13 | 8202 | 8282 | ||||||||||
14 | 138 | 716 | 180 | 1082 | 558 | |||||||
15 | 28 | 28 | 274 | 274 | ||||||||
16 | 3440 | 4700 | 5192 | 5756 | ||||||||
17 | 169 | 200 | 208 | 352 | 352 | |||||||
18 | 490 | 542 | 542 | 764 | 764 | |||||||
19 | 254 | 202 | 334 | 190 | 196 | |||||||
20 | 103 | 116 | 136 | 154 | ||||||||
21 | 388 | 459 | 454 | 558 | 354 | |||||||
22 | 1133 | 6246 | 1182 | 7440 | 2302 | |||||||
23 | 119 | 163 | 148 | 208 | 90 | |||||||
24 | 142 | 231 | 184 | 314 | 136 | 126 | ||||||
25 | 5654 | 8484 | 7526 | |||||||||
26 | ||||||||||||
27 | ||||||||||||
28 | 1837 | 2518 | 1624 | 1814 | 2108 | |||||||
29 | 2867 | 3058 | 3400 | |||||||||
30 | 204 | 4932 | 5623 | |||||||||
31 | 523 | 809 | 688 | 820 | 178 | |||||||
32 | 6511 | 8512 | ||||||||||
33 | 1439 | 1454 | 1240 | |||||||||
34 | 540 | 544 | 700 | 716 | ||||||||
35 | 1950 | 2231 | 2528 | 3058 | 1692 | |||||||
36 | ||||||||||||
37 | 384 | 413 | 486 | 564 | 214 | 268 | ||||||
38 | 17,061 | 3780 | ||||||||||
39 | 1366 | 2248 | ||||||||||
40 | 4002 | 3665 | 6146 | 5604 | 1254 | |||||||
41 | 1536 | 1655 | 2256 | 2456 | 1186 | |||||||
42 | 1740 | 2238 | 2476 | 3332 | 1138 | |||||||
43 | 432 | 570 | 226 | 766 | 642 | |||||||
44 | 143 | 112 | 190 | 106 | 118 | |||||||
45 | 364 | 987 | 392 | 1400 | 602 | |||||||
46 | 1043 | 1054 | 8742 | |||||||||
47 | 348 | 328 | 494 | 460 | 226 | |||||||
48 | 1141 | 1102 | 1484 | 1006 | 1134 | |||||||
49 | 5331 | 2452 | 6066 | |||||||||
50 | 1414 | 1312 | 1662 | 1322 | 1462 | |||||||
51 | 2965 | 10,470 | 3114 | 3122 | ||||||||
52 | 122 | 125 | 156 | 162 | 118 | |||||||
53 | 2805 | 2948 | 3710 | 3958 | 1858 | |||||||
54 | ||||||||||||
Average | ||||||||||||
Median |
Problem No. | BIRECT-(New) | BIRECT [26,41] | BIRECT-l-(New) | BIRECT-l [26] | ||||
---|---|---|---|---|---|---|---|---|
1 | 202 | 202 | 176 | 176 | ||||
2 | 1268 | 454 | 454 | |||||
3 | 874 | 874 | ||||||
4 | 436 | 436 | 436 | 436 | ||||
5 | 476 | 468 | 468 | |||||
6 | 472 | 478 | 472 | 472 | ||||
7 | 480 | 474 | 474 | |||||
8 | 194 | 188 | 188 | |||||
9 | 242 | 242 | 242 | 242 | ||||
10 | 794 | 794 | 794 | 794 | ||||
11 | 722 | 722 | 722 | 722 | ||||
12 | 4060 | 4060 | 4060 | 4060 | ||||
13 | ||||||||
14 | 16,420 | 480 | ||||||
15 | 274 | 274 | 274 | 274 | ||||
16 | 5106 | 5106 | ||||||
17 | 352 | 352 | 352 | 352 | ||||
18 | 764 | 764 | 764 | 764 | ||||
19 | 334 | 190 | 190 | |||||
20 | 152 | 152 | 152 | 152 | ||||
21 | 1024 | 660 | ||||||
22 | 7904 | 1698 | 1698 | |||||
23 | 94 | 90 | 90 | |||||
24 | 126 | 126 | 126 | 126 | ||||
25 | ||||||||
26 | ||||||||
27 | ||||||||
28 | 2114 | 2114 | 1832 | |||||
29 | ||||||||
30 | 4994 | |||||||
31 | 180 | 180 | 156 | |||||
32 | 1394 | 474 | ||||||
33 | 1250 | 1250 | ||||||
34 | 242 | 242 | 242 | 242 | ||||
35 | 1690 | 1700 | 1496 | |||||
36 | 4620 | |||||||
37 | 236 | 236 | 214 | |||||
38 | 7210 | 1422 | ||||||
39 | ||||||||
40 | 1272 | 1286 | ||||||
41 | 1204 | 1224 | 1224 | |||||
42 | 1140 | 1140 | 1162 | |||||
43 | 1780 | 1780 | 2114 | 2114 | ||||
44 | 118 | 118 | 108 | |||||
45 | 712 | 294 | ||||||
46 | 784 | 784 | ||||||
47 | 244 | 226 | 226 | |||||
48 | 1034 | 836 | 836 | |||||
49 | 7688 | 3366 | 3366 | |||||
50 | 1506 | 1138 | ||||||
51 | ||||||||
52 | 502 | 338 | 338 | |||||
53 | ||||||||
54 | ||||||||
Average | ||||||||
Median |
Problem No./Δ | BIRECT-l | |||||
---|---|---|---|---|---|---|
10−2 | 10−3 | 10−4 | 10−5 | 10−6 | 10−7 | |
1 | 168 | 182 | 178 | 174 | 176 | |
2 | 530 | 448 | 484 | 448 | 454 | 454 |
3 | 840 | 842 | 852 | 874 | 872 | 874 |
4 | 370 | 424 | 434 | 436 | 436 | 436 |
5 | 328 | 424 | 456 | 468 | 468 | 468 |
6 | 328 | 432 | 462 | 472 | 472 | 472 |
7 | 942 | 474 | 474 | 474 | ||
8 | 172 | 188 | 188 | 188 | 188 | 188 |
9 | 256 | 242 | 242 | 242 | 242 | |
10 | 722 | 790 | 794 | 790 | 794 | 794 |
11 | 732 | 718 | 722 | 722 | 722 | |
12 | 5352 | 4038 | 4060 | 4060 | 4060 | |
13 | ||||||
14 | 110 | 110 | 110 | 110 | 110 | 110 |
15 | 236 | 272 | 274 | 274 | 274 | 274 |
16 | 3236 | 3452 | 4148 | 4982 | ||
17 | 346 | 354 | 352 | 352 | 352 | 352 |
18 | 752 | 764 | 764 | 764 | 764 | 764 |
19 | 188 | 190 | 190 | 190 | 190 | 190 |
20 | 136 | 152 | 152 | 152 | 152 | 152 |
21 | 608 | 644 | 656 | 656 | 656 | 656 |
22 | 1590 | 1698 | 1698 | 1698 | 1698 | 1698 |
23 | 88 | 90 | 90 | 90 | 90 | 90 |
24 | 110 | 126 | 126 | 126 | 126 | 126 |
25 | ||||||
26 | ||||||
27 | ||||||
28 | 2440 | 2102 | 1814 | 1820 | 1820 | 1820 |
29 | ||||||
30 | 4810 | 5024 | 5014 | 5002 | 4994 | |
31 | 146 | 152 | 154 | 156 | 156 | 156 |
32 | 338 | 436 | 474 | 474 | 474 | 474 |
33 | 940 | 1166 | 1240 | 1250 | 1250 | 1250 |
34 | 236 | 242 | 242 | 242 | 242 | 242 |
35 | 1390 | 1470 | 1498 | 1496 | 1496 | 1496 |
36 | 4196 | 4510 | 4612 | 4620 | 4620 | 4620 |
37 | 172 | 204 | 214 | 214 | 214 | 214 |
38 | 1148 | 1280 | 1400 | 1434 | 1434 | 1074 |
39 | ||||||
40 | 666 | 810 | 1254 | 1248 | 1248 | 1248 |
41 | 636 | 818 | 1186 | 1224 | 1224 | 1224 |
42 | 632 | 766 | 1138 | 1162 | 1162 | 1162 |
43 | 1748 | 1880 | 2044 | 2086 | 2114 | 2114 |
44 | 104 | 106 | 106 | 106 | 106 | 106 |
45 | 286 | 294 | 294 | 294 | 294 | 294 |
46 | 786 | 784 | 784 | 784 | 784 | 784 |
47 | 202 | 226 | 226 | 226 | 226 | 226 |
48 | 728 | 826 | 836 | 836 | 836 | 836 |
49 | 2712 | 3162 | 3332 | 3366 | 3366 | 3366 |
50 | 1152 | 988 | 992 | 992 | 992 | 992 |
51 | ||||||
52 | 260 | 320 | 338 | 338 | 338 | 338 |
53 | ||||||
54 | ||||||
Average | ||||||
Median |
Algorithm | BIRECTv-l(impr.) | BIRECTv(impr.) | BIRECTv-l | BIRECTv | BIRECT-l | BIRECT |
---|---|---|---|---|---|---|
Success | ||||||
Fails | ||||||
max f.eval. | ||||||
min f.eval. | 28 | 28 | 80 | 80 | ||
average f.eval. | ||||||
Standard Deviation (std) | ||||||
median f.eval. |
Algorithm | |||||
---|---|---|---|---|---|
BIRECTv-l(impr.) | 2.313 | 2.321 | 2.398 | 2.344 | 2.327 |
BIRECTv(impr.) | 3.594 | 3.641 | 3.636 | 3.531 | 3.541 |
BIRECTv-l | 4.031 | 3.987 | 4.011 | 3.938 | 3.888 |
BIRECTv | 5.250 | 5.282 | 5.273 | 5.229 | 5.224 |
BIRECT-l | 2.594 | 2.513 | 2.432 | 2.563 | 2.571 |
BIRECT | 3.219 | 3.256 | 3.250 | 3.396 | 3.449 |
p-value |
Algorithm | |||||
---|---|---|---|---|---|
BIRECTv(impr.) | |||||
BIRECTv-l | |||||
BIRECTv | |||||
BIRECT-l | |||||
BIRECT |
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Belkacem, N.-E.; Chiter, L.; Louaked, M. A Novel Approach to Enhance DIRECT-Type Algorithms for Hyper-Rectangle Identification. Mathematics 2024, 12, 283. https://doi.org/10.3390/math12020283
Belkacem N-E, Chiter L, Louaked M. A Novel Approach to Enhance DIRECT-Type Algorithms for Hyper-Rectangle Identification. Mathematics. 2024; 12(2):283. https://doi.org/10.3390/math12020283
Chicago/Turabian StyleBelkacem, Nazih-Eddine, Lakhdar Chiter, and Mohammed Louaked. 2024. "A Novel Approach to Enhance DIRECT-Type Algorithms for Hyper-Rectangle Identification" Mathematics 12, no. 2: 283. https://doi.org/10.3390/math12020283
APA StyleBelkacem, N.-E., Chiter, L., & Louaked, M. (2024). A Novel Approach to Enhance DIRECT-Type Algorithms for Hyper-Rectangle Identification. Mathematics, 12(2), 283. https://doi.org/10.3390/math12020283