High-dimensional Multi-objective Optimization Algorithm Based on Large Language Model Assistance

doi:10.16182/j.issn1004731x.joss.25-1017

Abstract

Abstract:

The increase in the dimension of objective functions in optimization problems makes it challenging to balance convergence and diversity in population-based algorithms, which is a key factor leading to performance degradation of multi-objective evolutionary algorithms. To address this issue, a dynamic switching mechanism assisted by a large language model is proposed. By continuously evaluating the changes in hypervolume and convergence metrics of the solution set, the search process is adaptively divided into three phases: convergence-oriented, diversity-oriented, and large language model-suggested phases, achieving precise allocation of computational resources. Furthermore, to mitigate the high computational cost of large language models in high-dimensional scenarios, a low-cost invocation strategy is designed: the model is invoked only when performance declines for several consecutive generations, thereby guiding the population toward more efficient search. Comparative experiments with six state-of-the-art algorithms are conducted on 5 to 15-objective test sets from the DTLZ and WFG benchmark problems. The results demonstrate that the proposed method exhibits superior performance in both convergence and diversity metrics, confirming the effectiveness of large language models in assisting evolutionary algorithms to handle complex high-dimensional optimization problems.

Key words: high-dimensional multi-objective optimization, evolutionary algorithm, co-evolution, dynamic switching mechanism, large language model

CLC Number:

TP391.9

Ma Yichao, Wei Jingxuan. High-dimensional Multi-objective Optimization Algorithm Based on Large Language Model Assistance[J]. Journal of System Simulation, 2026, 38(9): 2574-2589.

Figures/Tables 12

Fig. 1

Fig. 2

Fig. 3

Table 1

Experimental results of IGD metrics for seven algorithms on the DTLZ test suite

Problem

NRVMOEA

HEA

TS-NSGA-II

MaOEA-IGD

RVEA

SSCEA

SSCEA-LLM

DTLZ1

7.276 9e-2

(4.21e-3) -

6.398 0e-2

(5.73e-4) -

6.516 4e-2

(2.77e-3) -

5.346 8e-1

(5.33e-1) -

6.329 8e-2

(4.00e-4) -

5.715 9e-2

(2.04e-2) =

5.328 5e-2

(1.23e-3)

7.361 4e+0

(2.30e+0) -

1.131 0e-1

(8.42e-4) =

1.843 5e-1

(3.65e-2) -

5.852 2e-1

(4.53e-1) -

1.125 1e-1

(1.02e-3) =

1.150 1e-1

(5.88e-3) =

1.140 5e-1

(6.17e-3)

1.131 0e+1

(2.98e+0) -

1.937 3e-1

(4.21e-3) -

2.528 0e-1

(2.62e-2) -

4.071 9e-1

(2.58e-1) -

1.743 4e-1

(5.93e-3) -

1.514 4e-1

(1.46e-2) =

1.627 2e-1

(4.23e-2)

DTLZ2

2.016 3e-1

(1.13e-3) -

1.948 4e-1

(5.82e-5) -

1.951 0e-1

(2.23e-4) -

2.014 1e-1

(1.47e-3) -

1.949 0e-1

(1.73e-5) -

1.680 5e-1

(1.07e-3) -

1.666 1e-1

(1.18e-3)

5.022 9e-1

(6.21e-3) -

4.393 5e-1

(1.13e-3) -

4.486 4e-1

(4.19e-3) -

6.425 4e-1

(1.38e-1) -

4.374 0e-1

(2.14e-4) -

4.137 8e-1

(3.54e-3) -

4.106 0e-1

(4.17e-3)

6.383 5e-1

(1.60e-2) -

6.301 3e-1

(2.29e-4) -

6.310 2e-1

(4.56e-3) -

8.579 9e-1

(8.59e-2) -

6.258 4e-1

(6.01e-3) -

5.253 1e-1

(4.44e-3) -

5.219 6e-1

(6.73e-3)

DTLZ3

6.694 0e-1

(6.23e-1) -

1.987 6e-1

(4.28e-3) =

3.947 1e-1

(3.99e-1) =

1.657 6e+1

(7.40e+0) -

4.180 2e-1

(4.04e-1) =

7.089 4e-1

(6.93e-1) =

4.653 5e-1

(3.69e-1)

4.093 8e+1

(1.37e+1) -

4.435 6e-1

(2.92e-3) +

8.967 0e-1

(1.82e-1) -

8.983 2e+0

(7.06e+0) -

4.449 7e-1

(2.58e-2) =

4.914 6e-1

(1.43e-1) -

4.480 0e-1

(4.09e-2)

8.456 4e+1

(2.53e+1)-

6.425 9e-1

(3.83e-2) -

1.388 6es+0

(6.31e-1) -

5.745 6e+0

(2.67e+0) -

6.343 0e-1

(1.17e-2) +

6.649 3e-1

(1.39e-1) -

6.383 4e-1

(1.69e-1)

DTLZ4

2.439 6e-1

(7.82e-2) -

3.153 3e-1

(1.65e-1) -

1.961 6e-1

(3.26e-4) -

2.836 4e-1

(1.54e-1) -

1.949 0e-1

(3.97e-5) -

1.852 3e-1

(5.86e-2) =

1.701 1e-1

(1.09e-3)

5.079 9e-1

(2.05e-2) -

4.737 2e-1

(7.65e-2) -

4.546 8e-1

(9.91e-3) -

5.097 3e-1

(5.96e-2) -

4.564 5e-1

(3.46e-2) -

4.306 4e-1

(1.40e-3) -

4.293 2e-1

(2.13e-3)

6.167 7e-1

(1.09e-2) -

6.383 2e-1

(9.67e-3) -

6.357 7e-1

(9.75e-3) -

6.470 7e-1

(1.49e-2) -

6.308 9e-1

(6.50e-3) -

5.485 9e-1

(1.46e-3) -

5.475 5e-1

(2.07e-3)

DTLZ5

9.594 5e-2

(1.45e-2) -

8.905 9e-2

(1.65e-2) -

1.565 7e-1

(3.65e-2) -

5.198 1e-1

(1.49e-1) -

1.785 4e-1

(2.63e-2) -

5.554 5e-2

(8.09e-3) -

5.021 1e-2

(8.45e-3)

2.413 7e-1

(6.79e-2) -

1.957 0e-1

(4.03e-2) -

4.658 9e-1

(2.25e-1) -

6.726 7e-1

(1.21e-1) -

5.473 1e-1

(2.22e-1) -

1.342 9e-1

(2.90e-2) -

1.167 8e-1

(2.20e-2)

3.576 6e-1

(5.24e-2) -

1.357 8e-1

(2.02e-2) +

3.827 7e-1

(8.01e-2) -

5.558 4e-1

(2.04e-1) -

4.971 8e-1

(2.22e-1) -

1.750 3e-1

(3.71e-2) =

1.580 3e-1

(2.96e-2)

DTLZ6

2.078 4e-1

(5.42e-2) -

1.990 7e-1

(6.40e-2) -

2.490 1e-1

(1.11e-1) -

7.328 9e-1

(2.21e-1) -

1.386 3e-1

(2.83e-2) -

7.805 0e-2

(2.06e-2) -

6.064 9e-2

(1.13e-2)

4.431 5e-1

(7.01e-2) -

4.666 9e-1

(1.33e-1) -

4.186 0e-1

(1.82e-1) -

7.626 0e-1

(1.97e-1) -

2.618 9e-1

(9.19e-2) -

1.553 4e-1

(3.64e-2) =

1.507 3e-1

(2.20e-2)

8.751 6e-1

(4.14e-1) -

3.908 8e-1

(7.81e-2) -

8.744 0e-1

(1.00e+0) -

6.929 0e-1

(1.00e-1) -

2.065 9e-1

(4.31e-2) -

1.770 7e-1

(4.55e-2) =

1.817 3e-1

(3.95e-2)

DTLZ7

2.899 8e-1

(4.32e-2) -

6.029 9e-1

(3.11e-1) -

3.129 9e-1

(6.84e-3) -

7.010 8e-1

(2.26e-2) -

5.729 1e-1

(1.91e-2) -

2.656 2e-1

(6.51e-3) =

2.666 0e-1

(8.47e-3)

8.677 5e-1

(2.33e-2) =

2.192 3e+0

(1.25e+0) -

1.769 0e+0

(8.11e-2) -

1.494 5e+0

(1.40e-1) -

2.238 7e+0

(3.91e-1) -

8.815 3e-1

(1.04e-1) -

8.329 1e-1

(7.78e-2)

1.522 5e+0

(4.13e-2) -

6.910 1e+0

(3.86e-1) -

8.422 2e+0

(3.22e-1) -

2.739 8e+0

(1.37e-1) -

2.586 0e+0

(4.91e-1) -

1.499 1e+0

(4.16e-2) =

1.498 5e+0

(2.95e-2)

+/-/=

0/20/1

2/17/2

0/20/1

0/21/0

1/17/3

0/11/10

Table 1

Fig. 4

Table 2

Experimental results of the IGD metric for seven algorithms on the WFG test suite

Problem

NRVMOEA

HEA

TS-NSGA-II

MaOEA-IGD

RVEA

SSCEA

SSCEA-LLM

WFG1

4.976 9e-1

(3.07e-2) -

4.592 5e-1

(5.42e-2) =

4.719 5e-1

(3.09e-2) =

3.383 3e+0

(1.50e+0) -

4.647 4e-1

(2.27e-2) =

6.056 6e-1

(1.88e-1) -

4.610 2e-1

(7.72e-2)

1.214 6e+0

(5.76e-2) -

1.603 9e+0

(1.81e-1) -

1.313 0e+0

(9.11e-2) -

9.008 1e+0

(3.16e+0) -

1.184 0e+0

(6.55e-2) -

1.074 2e+0

(1.43e-1) -

9.931 0e-1

(2.97e-2)

1.832 3e+0

(1.05e-1) -

2.116 9e+0

(4.23e-1) -

1.639 8e+0

(4.74e-2) -

8.730 0e+0

(5.96e+0) -

1.807 7e+0

(6.26e-2) -

1.667 1e+0

(1.15e-1) -

1.567 6e+0

(5.04e-2)

WFG2

5.718 6e-1

(1.07e-2) -

4.599 0e-1

(6.76e-3) -

4.840 5e-1

(1.57e-2) -

1.635 1e+0

(4.30e-1) -

4.516 4e-1

(1.26e-2) -

3.901 6e-1

(4.77e-3) =

3.888 2e-1

(6.63e-3)

1.482 7e+0

(8.58e-2) -

1.410 1e+0

(2.04e-1) -

1.383 1e+0

(8.83e-2) -

3.018 1e+0

(6.44e-1) -

1.353 3e+0

(6.80e-2) -

1.071 7e+0

(2.44e-2) =

1.073 7e+0

(2.35e-2)

2.093 8e+0

(1.18e-1) -

2.008 0e+0

(3.12e-1) -

1.779 4e+0

(6.80e-2) -

5.748 4e+0

(6.32e+0) -

1.828 2e+0

(1.03e-1) -

1.657 3e+0

(4.50e-2) =

1.647 1e+0

(3.94e-2)

WFG3

5.766 5e-1

(6.24e-2) -

4.032 9e-1

(2.28e-2) +

5.998 1e-1

(9.39e-2) -

5.422 2e+0

(2.98e-2) -

5.675 4e-1

(3.55e-2) -

4.159 7e-1

(4.13e-2) +

4.489 9e-1

(4.05e-2)

3.053 2e+0

(3.95e-1) -

2.477 1e+0

(1.31e-1) -

3.379 5e+0

(3.60e-1) -

1.008 9e+1

(2.85e+0) -

4.019 8e+0

(9.77e-1) -

1.316 8e+0

(1.56e-1) +

1.539 4e+0

(1.49e-1)

6.947 1e+0

(1.34e+0) -

2.298 4e+0

(1.65e-1) =

4.888 6e+0

(3.99e-1) -

1.193 3e+1

(5.87e+0) -

7.420 8e+0

(1.64e+0) -

1.793 6e+0

(3.27e-1) +

2.183 9e+0

(3.75e-1)

WFG4

1.186 0e+0

(1.96e-2) -

1.173 9e+0

(1.56e-3) -

1.172 2e+0

(1.72e-3) -

6.515 8e+0

(1.17e+0) -

1.172 5e+0

(1.78e-3) -

9.650 0e-1

(6.31e-3) -

9.5831e-1

(7.33 e-3)

5.763 8e+0

(1.85e-1) -

5.086 5e+0

(2.78e-2) -

5.066 7e+0

(3.79e-2) -

1.322 8e+1

(2.83e+0) -

4.878 1e+0

(4.90e-2) -

4.244 2e+0

(2.91e-2) =

4.240 5e+0

(3.78e-2)

9.506 4e+0

(2.23e-1) -

9.445 2e+0

(6.62e-3) -

9.427 7e+0

(1.66e-2) -

2.300 1e+1

(4.35e+0) -

9.272 4e+0

(1.30e-1) -

7.868 8e+0

(5.27e-2) =

7.892 8e+0

(6.86e-2)

WFG5

1.160 7e+0

(1.54e-2) -

1.161 7e+0

(1.38e-3) -

1.159 4e+0

(1.61e-3) -

4.081 2e+0

(2.94e+0) -

1.162 4e+0

(1.69e-3) -

9.495 8e-1

(8.38e-3) -

9.459 4e-1

(1.14e-2)

5.794 0e+0

(1.37e-1) -

5.084 2e+0

(1.34e-2) -

5.083 5e+0

(1.34e-2) -

1.738 6e+1

(1.98e+0) -

4.809 7e+0

(2.63e-2) -

4.239 2e+0

(3.80e-2) -

4.196 2e+0

(3.65e-2)

1.001 1e+1

(2.16e-1) -

9.369 9e+0

(1.39e-2) -

9.385 3e+0

(1.26e-2) -

2.753 0e+1

(5.95e+0) -

9.179 1e+0

(6.68e-2) -

7.472 7e+0

(7.28e-2) +

7.657 1e+0

(1.16e-1)

WFG6

1.179 7e+0

(1.66e-2) -

1.162 8e+0

(2.57e-3) -

1.158 3e+0

(2.19e-3) -

5.208 1e+0

(1.50e+0) -

1.164 3e+0

(1.33e-3) -

9.674 6e-1

(5.90e-3) =

9.737 2e-1

(2.16e-2)

6.053 9e+0

(2.03e-1) -

5.129 8e+0

(2.84e-2) -

5.105 5e+0

(2.90e-2) -

1.265 9e+1

(5.64e+0) -

5.408 7e+0

(3.29e-1) -

4.305 2e+0

(5.82e-2) -

4.225 8e+0

(4.42e-2)

1.053 9e+1

(2.31e-1) -

9.402 9e+0

(1.63e-2) -

9.403 5e+0

(1.80e-2) -

1.768 8e+1

(9.18e+0) -

9.670 4e+0

(3.61e-1) -

7.588 7e+0

(7.56e-2) +

7.919 3e+0

(2.46e-1)

WFG7

1.193 6e+0

(2.30e-2) -

1.174 3e+0

(1.29e-3) -

1.170 6e+0

(2.65e-3) -

5.749 6e+0

(9.07e-1) -

1.176 1e+0

(1.74e-3) -

9.912 5e-1

(1.34e-2) =

9.934 0e-1

(1.66e-2)

6.080 0e+0

(1.38e-1) -

5.097 0e+0

(2.87e-2) -

5.074 2e+0

(5.20e-2) -

1.457 3e+1

(2.13e+0) -

4.983 9e+0

(9.53e-2) -

4.254 4e+0

(5.90e-2) -

4.225 7e+0

(2.93e-2)

9.913 5e+0

(1.95e-1) -

9.432 7e+0

(1.42e-2) -

9.404 4e+0

(2.85e-2) -

2.430 6e+1

(2.70e+0) -

9.310 1e+0

(1.65e-1) -

8.368 0e+0

(3.12e-1) =

8.315 3e+0

(3.53e-1)

WFG8

1.176 3e+0

(1.22e-2) -

1.165 7e+0

(2.14e-3) -

1.166 9e+0

(2.55e-3) -

4.666 2e+0

(9.62e-1) -

1.169 3e+0

(1.51e-3) -

1.044 9e+0

(9.48e-3) =

1.053 4e+0

(1.91e-2)

5.973 5e+0

(1.32e-1) -

5.350 2e+0

(5.14e-2) -

5.218 8e+0

(2.89e-2) -

1.583 2e+1

(3.47e+0) -

5.485 3e+0

(2.31e-1) -

4.384 3e+0

(8.56e-2) =

4.371 6e+0

(9.17e-2)

1.042 3e+1

(2.82e-1) -

9.378 7e+0

(1.10e-2) =

9.326 6e+0

(2.53e-2) =

2.450 1e+1

(2.94e+0) -

9.549 6e+0

(4.35e-1) =

7.801 1e+0

(2.45e-1) +

9.254 1e+0

(8.39e-1)

WFG9

1.113 0e+0

(1.45e-2) -

1.130 6e+0

(4.81e-3) -

1.122 2e+0

(7.51e-3) -

4.796 0e+0

(1.18e+0) -

1.140 1e+0

(3.31e-3) -

9.845 6e-1

(1.84e-2) =

9.850 3e-1

(2.17e-2)

5.355 3e+0

(1.28e-1) -

5.102 5e+0

(1.30e-2) -

5.112 3e+0

(4.48e-2) -

1.395 1e+1

(4.21e+0) -

4.835 0e+0

(5.11e-2) -

4.302 3e+0

(4.79e-2) -

4.273 7e+0

(4.40e-2)

9.133 3e+0

(1.59e-1) -

9.154 6e+0

(2.26e-2) -

8.858 9e+0

(1.17e-1) -

2.282 6e+1

(6.86e+0) -

9.190 0e+0

(1.36e-1) -

7.964 4e+0

(1.56e-1) -

7.838 8e+0

(3.11e-1)

+/-/=

0/27/0

1/23/3

0/25/2

0/27/0

0/25/2

6/10/11

Table 2

Fig. 5

Table 3

Table 4

Table 5

Table 6

Table 7

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Problem	M	D	SSCEA-RND	SSCEA-CHG	SSCEA-LLM
DTLZ1	5	9	5.546 8e-2 (5.55e-3) =	5.470 7e-2 (1.93e-3) -	5.328 5e-2 (1.23e-3)
	10	14	1.124 3e-1 (5.02e-3) =	1.173 3e-1 (7.37e-3) =	1.140 5e-1 (6.17e-3)
	12	16	1.373 7e-1 (3.14e-2) =	1.294 9e-1 (1.01e-2) =	1.256 9e-1 (9.60e-3)
DTLZ2	5	14	1.680 0e-1 (3.66e-3) -	1.680 8e-1 (1.21e-3) -	1.666 1e-1 (1.18e-3)
	10	19	4.124 0e-1 (7.03e-3) =	4.158 7e-1 (3.00e-3) -	4.106 0e-1 (4.17e-3)
	12	21	4.737 3e-1 (4.88e-3) -	4.705 4e-1 (6.82e-3) -	4.667 1e-1 (5.57e-3)
DTLZ3	5	14	6.783 3e+0 (4.77e+0) -	2.547 8e+0 (1.77e+0) -	4.653 5e-1 (3.69e-1)
	10	19	1.694 8e+0 (1.33e+0) -	5.691 1e-1 (2.67e-1) -	4.480 0e-1 (4.09e-2)
	12	21	1.764 5e+0 (1.37e+0) -	7.209 8e-1 (2.92e-1) -	5.701 7e-1 (1.68e-1)
DTLZ4	5	14	1.700 2e-1 (1.22e-3) =	1.696 1e-1 (9.61e-4) +	1.701 1e-1 (1.09e-3)
	10	19	4.281 6e-1 (2.62e-3) =	4.295 1e-1 (1.93e-3) =	4.293 2e-1 (2.13e-3)
	12	21	4.944 7e-1 (2.12e-3) =	4.948 5e-1 (2.17e-3) =	4.955 3e-1 (2.10e-3)
DTLZ5	5	14	5.464 2e-2 (1.07e-2) =	5.330 5e-2 (6.62e-3) -	5.021 1e-2 (8.45e-3)
	10	19	1.303 8e-1 (2.69e-2) -	1.271 7e-1 (3.49e-2) =	1.167 8e-1 (2.20e-2)
	12	21	1.435 3e-1 (3.53e-2) =	1.614 7e-1 (4.57e-2) =	1.450 7e-1 (3.10e-2)
DTLZ6	5	14	6.497 9e-2 (1.25e-2) =	6.361 9e-2 (1.34e-2) =	6.064 9e-2 (1.13e-2)
	10	19	1.776 3e-1 (4.88e-2) -	1.571 9e-1 (2.79e-2) =	1.507 3e-1 (2.20e-2)
	12	21	1.718 2e-1 (3.99e-2) =	1.787 8e-1 (4.35e-2) =	1.740 2e-1 (6.55e-2)
DTLZ7	5	24	2.649 3e-1 (8.57e-3) =	2.688 0e-1 (9.50e-3) =	2.666 0e-1 (8.47e-3)
	10	29	8.037 9e-1 (4.79e-2) =	8.418 4e-1 (9.72e-2) =	8.329 1e-1 (7.78e-2)
	12	31	9.870 6e-1 (3.01e-2) -	9.574 4e-1 (1.70e-2) =	9.650 0e-1 (3.75e-2)
+/-/=			0/8/13	1/8/12

Problem	M	D	SSCEA-LLM-5	SSCEA-LLM-1	SSCEA-LLM-3
+/-/=			0/2/5	0/2/5
DTLZ1	5	9	6.060 2e-2 (1.80e-2) -	6.357 8e-2 (1.17e-2) -	5.324 8e-2 (1.46e-3)
DTLZ2	5	14	1.677 4e-1 (3.66e-3) =	1.670 6e-1 (1.06e-3) =	1.667 9e-1 (1.15e-3)
DTLZ3	5	14	8.759 7e-1 (8.03e-1) =	2.555 9e-1 (1.06e-1) =	5.221 6e-1 (3.95e-1)
DTLZ4	5	14	1.704 7e-1 (1.40e-3) =	1.702 4e-1 (1.14e-3) =	1.700 1e-1 (1.16e-3)
DTLZ5	5	14	5.824 2e-2 (1.05e-2) -	5.859 9e-2 (9.29e-3) -	5.018 1e-2 (5.41e-3)
DTLZ6	5	14	6.693 5e-2 (1.51e-2) =	6.569 3e-2 (1.65e-2) =	6.142 8e-2 (1.03e-2)
DTLZ7	5	24	2.668 8e-1 (6.34e-3) =	2.776 1e-1 (4.36e-2) =	2.682 3e-1 (8.26e-3)

Problem	M	D	SSCEA-LLM-5	SSCEA-LLM-1	SSCEA-LLM-3
+/-/=			7/0/0	0/6/1
DTLZ1	5	9	3.171 9e+1 (5.91e+0) +	2.186 7e+2 (2.09e+1) -	6.016 8e+1 (9.68e+0)
DTLZ2	5	14	2.577 8e+1 (4.05e+0) +	1.837 4e+2 (2.48e+1) -	4.730 2e+1 (1.12e+1)
DTLZ3	5	14	4.569 4e+1 (6.90e+0) +	2.786 2e+2 (2.69e+1) =	5.108 3e+2 (3.73e+2)
DTLZ4	5	14	2.222 5e+1 (4.70e+0) +	1.986 8e+2 (2.01e+1) -	4.638 8e+1 (1.46e+1)
DTLZ5	5	14	1.838 7e+1 (3.85e+0) +	1.729 4e+2 (1.61e+1) -	3.183 2e+1 (1.06e+1)
DTLZ6	5	14	2.221 5e+1 (3.29e+0) +	1.984 9e+2 (1.39e+1) -	4.674 8e+1 (7.86e+0)
DTLZ7	5	24	1.287 0e+1 (4.47e+0) +	1.702 1e+2 (1.27e+1) -	3.141 7e+1 (5.92e+0)

Algorithm	DTLZ1	DTLZ2	DTLZ3	WFG1	WFG2
MOEA-D-DQN	5.487 1e+1	5.657 2e+1	1.176 3e+2	1.052 9e+2	1.137 4e+2
RVEA-IGNG	2.472 3e+1	3.249 3e+1	2.035 3e+1	2.686 2e+1	3.267 9e+1
NRVMOEA	1.500 6e+1	1.556 3e+1	1.497 7e+1	1.523 7e+1	1.588 6e+1
SSCEA	5.016 2e+0	5.363 8e+0	4.867 1e+0	5.659 8e+0	5.071 6e+0
MOEAD-LLM	N/A	N/A	N/A	N/A	N/A
SSCEA-LLM	5.311 8e+1	4.455 7e+1	7.248 3e+1	4.839 7e+1	3.939 6e+1