受约束资源竞争的二阶惯性边流竞争模型: 守恒性、KKT平衡与有界优势集聚

黄毅; 朱扬威; 陈祖国; 张志鑫; 黄廷文

doi:10.16383/j.aas.c260212

受约束资源竞争的二阶惯性边流竞争模型: 守恒性、KKT平衡与有界优势集聚

doi: 10.16383/j.aas.c260212 cstr: 32138.14.j.aas.c260212

黄毅^{1, 2,},
朱扬威^{1, 2,},
陈祖国^{1, 2,},
张志鑫^3,,
黄廷文^4,

1.
湖南科技大学三亚研究院三亚 572024
2.
湖南科技大学信息与电气工程学院湘潭 411201
3.
华东交通大学电气与自动化工程学院江西 330013
4.
深圳理工大学计算机科学与人工智能学院深圳 518107

基金项目: 国家自然科学基金项目(62403193, 62373144), 湖南省科技创新项目(2025RC3197, 2024RC9015), 海南省自然科学基金项目(626MS0242, 626MS0243)资助

详细信息

作者简介:
黄毅：湖南科技大学信息与电气工程学院特聘副教授. 主要研究方向为复杂系统建模, 控制与优化, 强化学习. 本文通信作者. E-mail: yi.huang@hnust.edu.cn

朱扬威：湖南科技大学信息与电气工程学院硕士研究生. 主要研究方向为建模与控制, 强化学习. E-mail: ywzhu2003@163.com

陈祖国：湖南科技大学信息与电气工程学院副教授. 主要研究方向为人工智能, 机器人控制技术和复杂工业过程智能决策. E-mail: zg.chen@hnust.edu.cn

张志鑫：华东交通大学电气与自动化工程学院讲师. 主要研究方向为非线性系统控制, 容错控制, 多智能体系统和瞬态性能约束控制. E-mail: zhixin.zhang@ecjtu.edu.cn

黄廷文：深圳理工大学计算机科学与人工智能学院教授. 主要研究方向为神经网络, 复杂网络, 混沌与系统动力学, 算子半群理论及其应用. E-mail: huangtingwen@suat-sz.edu.cn

计量
- 文章访问数: 16
- HTML全文浏览量: 10
- 被引次数: 0
出版历程
- 收稿日期: 2026-03-30
- 录用日期: 2026-06-29
- 网络出版日期: 2026-07-27

A Second-order Inertial Edge-flow Competition Model for Constrained Resource Competition: Conservation, KKT Equilibria, and Bounded Dominant Concentration

HUANG Yi^{1, 2
,},
Zhu Yang-Wei^{1, 2
,},
CHEN Zu-Guo^{1, 2
,},
ZHANG Zhi-Xin^3
,,
HUANG Ting-Wen^4
,

1.
Sanya Institute of Hunan University of Science and Technology, Sanya 572024
2.
School of Information and Electrical Engineering, Hunan University of Science and Technology, Xiangtan 411201
3.
School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang 330013
4.
Faculty of Computer Science and Artificial Intelligence, Shenzhen University of Advanced Technology, Shenzhen 518107

Funds: Supported by National Natural Science Foundation of China (62403193, 62373144), Science and Technology Innovation Program of Hunan Province (2025RC3197, 2024RC9015), and Natural Science Foundation of Hainan Province (626MS0242, 626MS0243)

More Information

Author Bio:
HUANG Yi　Distinguished associate professor at the School of Information and Electrical Engineering, Hunan University of Science and Technology. His research interests include complex systems modeling, control and optimization, and reinforcement learning. Corresponding author of this paper

ZHU Yang-Wei　Master student at the School of Information and Electrical Engineering, Hunan University of Science and Technology. His research interests include modeling and control, and reinforcement learning

CHEN Zu-Guo　Associate professor at the School of Information and Electrical Engineering, Hunan University of Science and Technology. His research interests include artificial intelligence, robot control technology, and intelligent decision-making for complex industrial processes

ZHANG Zhi-Xin　Lecturer at the School of Electrical and Automation Engineering, East China Jiaotong University. His research interests include nonlinear systems control, fault-tolerant control, multi-agent systems, and transient performance constrained control

HUANG Ting-Wen　Professor at the Faculty of Computer Science and Artificial Intelligence, Shenzhen University of Advanced Technology. His research interests include neural networks, complex networks, chaos and dynamics of systems, and operator semigroups theory and its applications

摘要

摘要: 针对现有一阶赢者通吃竞争模型难以同时刻画惯性效应、状态约束与严格优化解释, 本文研究资源总量守恒与箱体约束下的分布式非凸资源分配问题. 为此, 针对 Hill 型效用函数引入对数障碍正则化, 并以图边流动量为中间状态构建二阶惯性边流竞争模型. 所建模型能够利用局部边际效用信息实现分布式交互. 随后, 通过构造能量函数并运用 LaSalle 不变性原理, 证明可行内点集具有正不变性、系统能量单调耗散, 且系统轨迹收敛至平衡点集合. 在此基础上, 进一步建立动力学平衡点与正则化问题 KKT 条件之间的对应关系, 并证明当障碍参数趋于零且相关对偶变量保持有界时, 平衡点极限能够恢复原问题的 KKT 结构. 此外, 针对智能体处于 Hill 边际效用饱和区的情形, 给出能力参数与稳态资源分配之间的条件排序性质. 数值结果表明, 所建模型能够在保持约束的同时形成有界优势资源集聚. 其中, 惯性环节在收敛速度与峰值边流强度之间提供可调折中, 网络拓扑主要影响边际效用一致化速度, 而总资源规模则改变优势资源集聚强度.
- 二阶惯性边流资源竞争模型 /
- 非凸资源分配 /
- 资源守恒
Abstract: This paper studies distributed nonconvex resource allocation under total-resource conservation and box constraints, addressing the difficulty of existing first-order winner-take-all competition models cannot simultaneously characterize inertial effects, state constraints, and a rigorous optimization interpretation. To this end, logarithmic barrier regularization is introduced for the Hill-type utility function, and a second-order inertial edge-flow competition model is constructed by using graph edge-flow momentum as an intermediate state. The proposed model can realize distributed interactions based on local marginal utility information. Then, by constructing an energy function and applying LaSalle's invariance principle, it is proved that the feasible interior set is positively invariant, the system energy is monotonically dissipative, and the system trajectory converges to the equilibrium set. On this basis, the correspondence between the dynamical equilibria and the KKT conditions of the regularized problem is further established, and it is proved that, when the barrier parameter tends to zero and the associated dual variables remain bounded, the limiting equilibria can recover the KKT structure of the original problem. In addition, for the case where agents lie in the saturation region of the Hill marginal utility, a conditional ordering property between capability parameters and steady-state resource allocations is derived. Numerical results show that the proposed model can form bounded advantageous resource concentration while preserving the constraints. Specifically, the inertial term provides a tunable tradeoff between convergence speed and peak edge-flow intensity, the network topology mainly affects the marginal-utility consensus speed, and the total resource scale changes the intensity of advantageous resource concentration.
- second-order inertial edge-flow resource competition model /
- nonconvex resource allocation /
- resource conservation

HTML全文

图 1 固定障碍参数下的状态演化与约束保持

Fig. 1 State evolution and constraint preservation for a fixed barrier parameter

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图 2 固定障碍参数下的性能指标与收敛行为

Fig. 2 Performance metrics and convergence behavior for a fixed barrier parameter

下载: 全尺寸图片幻灯片

图 3 障碍参数扫描下的平衡分配与乘子趋势

Fig. 3 Equilibrium allocations and multiplier trends under a barrier-parameter sweep

下载: 全尺寸图片幻灯片

图 4 障碍参数趋零时的KKT残差验证

Fig. 4 Verification of KKT residuals as the barrier parameter approaches zero

下载: 全尺寸图片幻灯片

表 1 初值与稳态资源分配对比($ \mu=0.02 $)

Table 1 Comparison of initial and steady-state resource allocations ($ \mu=0.02 $)

智能体$ i $	能力参数$ c_i $	初始资源$ x_i(0) $	稳态资源$ x_i^\ast $
1	1.00	0.41	0.060 9
2	1.20	0.38	0.062 3
3	1.45	0.32	0.227 3
4	1.70	0.29	0.309 9
5	2.00	0.27	0.366 7
6	2.35	0.25	0.416 8
7	2.70	0.24	0.458 0
8	3.10	0.24	0.498 0

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表 3 随机初始条件和模型参数扰动下的统计结果

Table 3 Statistical results under random initial conditions and model-parameter perturbations

扰动类型	有效比例	$ J(x^\ast) $	$ t_s $	峰值边流	排序一致率
随机初值	1.00	$ 7.452\,10\pm0.010\,11 $	$ 1.97\pm0.28 $	$ 0.332\,19\pm0.080\,14 $	$ 1.000\pm0.000 $
能力参数组	1.00	$ 7.437\,15\pm0.256\,18 $	$ 2.78\pm1.07 $	$ 0.355\,16\pm0.034\,15 $	$ 1.000\pm0.000 $
Hill参数	1.00	$ 7.256\,15\pm1.653\,11 $	$ 2.87\pm1.30 $	$ 0.336\,19\pm0.035\,19 $	$ 1.000\pm0.000 $

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表 2 $ \mu $扫描下的平衡与残差指标

Table 2 Equilibrium and residual metrics under the $ \mu $ sweep

$ \mu $	$ \lambda^\ast(\mu) $	$ \max\|{\rm{stat}}\| $	$ \max\|{\rm{comp}}\| $	$ \max\|B^{{\rm{T}}} h_\mu\| $	$ \max(\mu_i^+,\mu_i^-) $
0.08	2.960 625	$ 6.61\times 10^{-10} $	$ 8.0\times 10^{-2} $	$ 8.17\times 10^{-10} $	1.700 7
0.04	2.809 298	$ 9.16\times 10^{-10} $	$ 4.0\times 10^{-2} $	$ 1.31\times 10^{-9} $	1.771 5
0.02	2.742 712	$ 1.45\times 10^{-9} $	$ 2.0\times 10^{-2} $	$ 2.29\times 10^{-9} $	1.831 0
0.01	2.710 407	$ 6.44\times 10^{-9} $	$ 1.0\times 10^{-2} $	$ 1.07\times 10^{-8} $	1.862 6
0.005	2.694 130	$ 1.73\times 10^{-8} $	$ 5.0\times 10^{-3} $	$ 2.94\times 10^{-8} $	1.878 3

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表 4 一阶/二阶模型与阻尼参数比较

Table 4 Comparison of the first- and second-order models under different damping parameters

模型	$ \alpha $	$ J(x^\ast) $	$ t_s $	峰值边流强度
一阶约化流	0	7.447 979	3.06	0
二阶惯性流	4	7.475 509	2.78	0.487 5
二阶惯性流	8	7.447 979	2.00	0.357 0
二阶惯性流	12	7.447 979	4.12	0.279 5
二阶惯性流	20	7.447 979	7.40	0.192 7

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表 5 不同网络拓扑和随机边权的平衡与敏感度分析

Table 5 Equilibrium and sensitivity analysis under different network topologies and random edge weights

拓扑	$ J(x^\ast) $	$ t_s $	活跃分量数	边际失配
链式图	7.447979	35.04	1	$ 2.19\times 10^{-10} $
环形图	7.447979	11.74	1	$ 3.04\times 10^{-9} $
加边环形图	7.447979	2.00	1	$ 1.16\times 10^{-9} $
完全图	7.447979	1.50	1	$ 6.04\times 10^{-9} $
星形图	7.447979	6.32	1	$ 6.46\times 10^{-9} $
随机边权加边环	7.447979	2.40 $ \pm $ 0.59	1	$ 1.73\times10^{-15} $

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表 6 总资源变化对优势集聚结构的影响

Table 6 Effect of total resource variation on the structure of advantageous resource concentration

$ C_{{\rm{total}}} $	$ J(x^\ast) $	下边界节点数	活跃节点数	前3节点资源占比
1.8	5.826 937	0	8	0.690 1
2.4	7.447 979	0	8	0.572 0
3.0	8.831 312	0	8	0.516 9
3.6	9.982 043	0	8	0.474 7

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表 7 20个智能体完全图稳态资源排序

Table 7 Steady-state resource ranking for complete graph of 20 agents

排名	节点$ i $	能力参数$ c_i $	稳态资源$ x_i^\ast $
1	20	3.400 0	0.389 7
2	19	3.273 7	0.377 6
3	18	3.147 4	0.364 8
4	17	3.021 1	0.351 1
5	16	2.894 7	0.336 3

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