
IEEE/CAA Journal of Automatica Sinica
Citation: | Reza Asadi and Solmaz S. Kia, "Cycle Flow Formulation of Optimal Network Flow Problems and Respective Distributed Solutions," IEEE/CAA J. Autom. Sinica, vol. 6, no. 5, pp. 1251-1260, Sept. 2019. doi: 10.1109/JAS.2019.1911705 |
IN a network flow problem, a physical system consisting of several routes between source and sink points transfers input flows from the source points to the sink points. The primary objective of optimal network flow problems is to minimize the overall cost of transporting flow [1]. Network flow problems appear in many important applications, such as software-defined networking [2], wireless sensor networks [3], transportation systems [4], [5], and power networks [6]–[8]. In power network problems, variants of optimal network flow problems also aim to include optimal generation and storage costs [9]–[12].
With the advent of new technologies, the amount of available data and network size has been increasing, which necessitate various performance improvement techniques in cyber-physical systems [13], [14], and increase the size of optimization problems. However, the number of decision variables is directly related with the time and space (resources①) computation complexity of optimization solvers. Decision variable reduction techniques, as well as parallel/distributed optimization solutions, are therefore investigated to manage the complexity of large-scale optimization problems. Distributed solutions are of particular interest in decentralized cyber-physical operations, which aim to solve network flow problems in a scalable, fast, and decentralized manner. Although the computational cost of distributed algorithms in an optimal network flow problem is distributed among the cyber-layer nodes, the high number of decision variables normally translates to a large number of cyber nodes or large in-network communication overhead. Additionally limitations of these solutions include network congestion and the direct effect of communication cost on life-expectancy of battery-operated cyber nodes. To address these limitations, our objective is to construct a variable reduction technique that reduces the size of the decision variables in optimal network flow problems, while maintaining a sparse structure amenable to distributed solutions at a lower communication cost.
Variable fixing techniques [15], dominance techniques [16], and constraint pairing techniques [17] are general variable reduction techniques in integer quadratic problems. Also, in multi-objective optimization problems data mining techniques are used to reduce less effective variables [18]. For evolutionary optimization problems, [19] presents how variable reduction techniques can be applied to obtain the variable relations from the partial derivatives of an optimization function. For optimization problems of the form (1), eliminating affine equality constraint, as discussed below, is also a method for reducing the number of the search variables of the problem (see [20])
x⋆=argminx∈Rmϕ(x),s.t.Ax=b;g(x)≤0, |
(1) |
where
The affine feasible set for this optimization problem can be characterized as
{x∈Rm∣Ax=b}={Fz+xp∣z∈Rm−ρ}, |
(2) |
where
z⋆=argminz∈Rn−ρˉϕ(z)=ϕ(Fz+xp),s.t.ˉg(z)=g(Fz+xp)≤0. |
(3) |
Compared to (1), in (3), the equality constraint is eliminated and the number of the decision variables are reduced from
Optimal network flow problems are normally in the form of the optimization problem (1), where the cost is the sum of the convex cost of the flow through the arcs subject to capacity bounds for each arc and flow conservation equations at each node. In variations of the optimal network flow problem, the cost can be augmented to include the cost of, e.g., generation and storage at nodes, and the constraints can be expanded to include other components. Nevertheless, an affine equality constraint that always is present in network flow problems is the flow conservation equation. The coefficient matrix of the linear equation describing the flow conservation constraints has a nullity of
Various matrix factorization techniques, including LU, QR, LQ, SVD, and Gauss-Jordan elimination are proposed to compute the span of the null-space of a matrix
Statement of Contribution: In this paper, we use a graph theoretic approach to reduce the decision variables of optimal network flow problems by eliminating the flow conservation constraint. In particular, we show that all solutions of the flow conservation equation are characterized explicitly in terms of the span of the columns of the transpose of a cycle basis matrix of the oriented network plus a particular solution. Cycle basis of a graph can be computed in polynomial time using efficient algorithms such as those in [28], [29]. To compute a particular solution, we then propose a graph theoretic approach. We show that for any given input/output flow vector, a particular solution can be efficiently constructed from a set of elementary solutions, each obtained from tracing a flow of value
Notations:
Organization of the Paper: Section II defines graph-related terminologies and some of the basic properties of the cycle basis method. Section III formally presents our decision variable reduction method and its application in a minimum cost flow and an optimal power flow problems. Section IV presents a distributed ADMM solution for the cycle-flow-formulated minimum cost network flow problem and discusses the communication reduction on some example networks. Section V demonstrates the performance of the distributed cycle basis ADMM on a numerical example in terms of convergence to optimum solution and the number of communications. Section VI provides the conclusions.
In this section, following [34], we review our graph-related terminology and conventions, and introduce our graph-related notations. We represent a graph of
When there is an orientation assigned to the arcs of a graph
A cycle of
Lemma 1 (Relationship between the oriented incidence matrix and an oriented cycle vector (see [34] for proof)): In an oriented graph
Next, note that the vector space over Q, the Galois field of 2, generated by the oriented cycle vectors is the cycle space of
In this section, we show how two well-known network flow problems can benefit from affine equality elimination method to reduce their search variables. We study our optimal network flow problems of interest over a network of
∑mj=1Ioijxj=fi,i∈{1,…,n}, |
(4) |
which in an aggregated form is represented with
Iox=f, |
(5) |
where
Theorem 1 (Solution set of (5)): Let
{x∈Rn|x=Bo⊤z+xp,z∈m−n+1}, |
(6) |
where
Proof: The proof relies on showing that the null-space of
In what follows, we use the result of Theorem 1 to develop an affine equality elimination method to reduce the decision variables of two optimal network flow problems. However, the effectiveness of this approach depends on how efficiently one can construct matrix
Next, we propose a simple method to construct a particular solution
Lemma 2: (Particular solution of (5)). Given an input/output flow vector
Proof: For every
A few remarks are in order regarding the particular solution. First, note that construction of the `fundamental' particular solution set
We consider a minimum cost flow problem over
x⋆=argminx∈Rmϕ(x)=∑mi=1ϕi(xi),s.t. |
(7a) |
∑mj=1Ioijxj=fi;i∈{1,…,n}; |
(7b) |
bj≤xj≤cj;j∈{1,⋯,m}, |
(7c) |
where
Theorem 2: (Eliminating the flow conservation constraint from (7)). Consider the optimal network flow problem (7) over a connected physical network
z⋆=argminz∈Rm−n+1ϕ(z)=m∑i=1ϕi(z⊤[Bo]i+xpi),s.t.bj≤z⊤[Bo]j+xpj≤cj,j∈{1,…,m}, |
(8) |
and
Proof: The proof follows from invoking the same argument that is used to relate solutions of the optimization problem (1) to those of (3), and the result of Theorem 1.■
Next, we consider an optimal power flow problem over a network described by
All of these models, at each time
In our study below, without loss of generality, we use the deterministic form (no renewable generation source) of the optimal network flow problem studied in [9], which states the problem as a direct current (DC) power flow problem (see (9)). Without loss of generality, let at each time
{x⋆(t),δ⋆(t),u⋆(t),s⋆(t),θ⋆(t)}Tt=1= |
(9a) |
argmin1T∑Tt=1(∑nj=1gj(δj(t))+∑mi=1ϕi(xi(t))), |
(9b) |
s.t.fort∈T,i∈{1,…,n}∑mj=1Ioijxj(t)=δi(t)−ui(t)+di(t); |
(9c) |
si(t+1)=λisi(t)+ui(t); |
(9d) |
Bik(θi(t)−θk(t))=xj(t);k∈Ne(i),ej=(vi,vk) |
(9e) |
δ_i≤δi(t)≤ˉδi;u_i≤ui(t)≤ˉui;s_i≤si(t)≤ˉsi; |
(9f) |
bj≤xj(t)≤cj;j∈{1,…,m}, |
(9g) |
where the cost function includes generator costs
The following result shows that the number of search variables related to the flow in the optimization problem (9) can be reduced from
Proposition 1: (Eliminating the flow conservation constraint from (9)). Consider the optimal power flow problem (9) over a physical network described by
{z⋆(t),δ⋆(t),u⋆(t),s⋆(t),θ⋆(t)}Tt=1=argmin1T∑Tt=1(∑nj=1gj(δj(t))+ |
(10a) |
∑mi=1ϕi(z(t)⊤[Bo]i+xpi(δ(t),u(t),d(t))),s.t.fort∈T,i∈{1,…,n}∑nj=1(δj(t)+uj(t)+dj(t))=0, |
(10b) |
si(t+1)=λisi(t)+ui(t), |
(10c) |
Bik(θi(t)−θk(t))=z(t)⊤[Bo]j+xpj(δ(t),u(t),d(t)),k∈Ne(i),ej=(vi,vk), |
(10d) |
δ_i≤δi(t)≤ˉδi,u_i≤ui(t)≤ˉui,s_i≤si(t)≤ˉsi, |
(10e) |
bj≤z(t)⊤[Bo]j+xpj(δ(t),u(t),d(t))≤cj,j∈{1,⋯,m}. |
(10f) |
Here,
Proof: The equality constraint (9c) in aggregated form is
Iox(t)=δ(t)+u(t)+d(t),t∈T. |
(11) |
Note that rank of
{x∈Rm∣Iox=δ+u+d,δ∈Rn,u∈Rn}={Bo⊤z+xp(δ,u,d)|xp(δ,u,d)=∑n−1i=1(δi+ui+di)ˉxp,vi,∑ni=1(δi+ui+di)=0,z∈Rn−m+1,δ∈Rn,u∈Rn}. |
As a result, at each
In this section, our objective is to demonstrate the effectiveness of the cycle basis decision variable reduction technique in decreasing the communication cost of distributed solutions of optimal network flow problems. We use the minimum network flow problem (7) as our demonstration case. For this problem, we first develop a distributed ADMM algorithm to solve (8). Then, we compare the communication cost of this algorithm to that of a distributed ADMM solution for (7).
To develop our results, we introduce first some notations related to the oriented cycles of
We let
Recall that to eliminate the flow conservation equation we used
ϕi(xi)=ϕi(z⊤[Bo]i+xpi)=ψi({zk}k∈Ic(ei)). |
Next, we derive equivalent representations of optimization problem (8) that can be solved in a distributed manner using the ADMM algorithm of [13] via two different cyber-layer architectures.
Cycle-Based Cyber-Layer: to develop our first distributed solution to solve (8), we assign a cyber-layer node to each cycle of the cycle basis that we used in our decision variable reduction stage (see Fig. 1 as an example). We refer to this architecture as cycle-based cyber-layer. We assume that the cyber-layer nodes of the neighboring cycles can communicate with each other in a bi-directional way. This procedure results in a cyber-layer with
Now, for every cyber-layer node
θi(yi)=∑∀ek∈ECi1|Ic(ei)|ψk(yi(ek)). |
(12) |
Then, we can cast the minimum cost network flow problem (8) in the following equivalent form
y⋆=argminy1,⋯,yμ∑μi=1θi(yi),s.t. |
Set of constraints at each cyber agent
{yi(ek)=[{ˉyj}j∈Ic(ei)],bk≤yi(ek)⊤[{Bojk}j∈Ic(ek)]+xpk≤ck,∀ek∈ECi. |
(13) |
In this formulation, every cycle-based cyber node has a copy of the cycle flows that go through its arcs, i.e,
One can solve also the original minimum cost network flow problem (7) in a distributed manner by a cycle-based cyber-layer using, for example, the distributed ADMM algorithm of [24]. To do so, every cyber-layer node needs a local copy of the arc flows across all the nodes in its corresponding cycle. The local copies are required to create a local copy of the flow conservation equation of the nodes. Then, to coordinate these local copies, the cyber-layer agents that share a node are required to communicate with each other. Since there are more cycles that are connected to each other through the nodes than those connected by arcs, as shown in Fig. 1, this distributed solution requires more connections in the cyber-layer than the distributed solution for (8). Moreover, because there are more arc flows than cycle flows, the message sizes exchanged among neighboring cyber agents solving (7) is larger. Therefore, the distributed solution for (7) will be less favorable from network congestion and communication energy consumption perspective.
Remark 1: (Example networks). In a square mesh physical layer network with
Node-Based Cyber-Layer: We can solve problem (8) in a distributed manner with the conventional node-based cyber-layer, as well. In a node-based cyber-layer architecture, a cyber agent is assigned to each node of the physical layer to compute the flow across all the incident arcs of the corresponding physical layer node. For example, in traffic networks, a cyber agent is assigned to each intersection, which is a node in the physical layer. In this case, the topology of the cyber-layer is exactly the same as the physical layer. To solve problem (8) using a distributed ADMM algorithm over such cyber-layer, we assume that each cyber agent has a local variable
θi(yi)=12∑∀ek∈JE(vi)ψk(yi(ek)),vi∈Vphysic. |
(14) |
Then, we can cast the minimum cost network flow problem (8) in the following equivalent form
y⋆=argminy1,⋯,yn∑ni=1θi(yi),s.t. |
Set of constraints at each cyber agent
{yi(ek)=yj(ek),ek=(vi,vj)∈Ephysic,bk≤yi(ek)⊤[{Bolk}l∈Ic(ek)]+xpk≤ck,∀ek∈JE(vi), |
(15) |
Given this equivalent representation, similar to the method described for the cycle-based cyber-layer, we can now solve the optimization problem (8) using a distributed ADMM algorithm. In a cycle flow representation, it is more likely to have fewer cycles that go through a node than the arcs that are incident at that node. Therefore, in a distributed ADMM solution, the size of the broadcast messages of a cyber agent solving (8) is more likely to be less than the size of the broadcast messages when we solve (7).
We demonstrate the use of distributed cycle-flow-based ADMM algorithm for a minimum cost optimal flow over the network shown in Fig. 1, and compare it with the arc-flow-based ADMM distributed algorithm [24]. The arrows in the physical layer show the positive flow orientation assigned to the arcs. The cyber-layer is generated based in the minimum weight cycle basis partitioning of the physical layer as shown in the bolder network with blue nodes in Fig. 1. In this problem, we set capacity bounds
Bo=[−110−10000000000000000−11000−11100000000000001−10100000000000000000000−110−10−11000000001−100001000000000000000000−11−1100000000000000000000−11−11] |
The central optimum solution of our example is
We have considered optimal network flow problems and investigated how the decision variables size of these problems can be reduced by eliminating the affine flow conservation equations. Our study was based on exploiting cycle basis concept from graph theory to eliminate the flow conservation equation in an efficient manner. In particular, we have shown that the components of our variable reduction method can be obtained in a systematic manner using graph theoretic approaches. Moreover, we have shown that the new formulation of the optimal network flow problems with reduced variables has a sparse structure and can be solved via distributed optimization solvers. In this regard, we have demonstrated the use of a distributed ADMM solver for the cycle-flow-based minimum cost flow formulation, and showed that this distributed operation leads to a reduced communication cost among the cyber-layer nodes.
[1] |
D. P. Bertsekas, Network Optimization: Continuous and Discrete Models. Citeseer, 1998.
|
[2] |
K. Qin, C. Huang, N. Ganesan, K. Liu, and X. Chen, " Minimum cost multi-path parallel transmission with delay constraint by extending openflow,” China Communications, vol. 15, no. 3, pp. 15–26, 2018. doi: 10.1109/CC.2018.8331988
|
[3] |
A. Sinha and E. Modiano, " Optimal control for generalized networkflow problems,” IEEE/ACM Transactions on Networking (TON)
|
[4] |
C. Rosdahl, G. Nilsson, and G. Como, " On distributed optimal control of traffic flows in transportation networks,” in Proc. IEEE Conf. on Control Technology and Applications, pp. 903–908, 2018.
|
[5] |
S. Pourazarm and C. G. Cassandras, " Optimal routing of energyaware vehicles in transportation networks with inhomogeneous charging nodes,” IEEE Transactions on Intelligent Transportation Systems, vol. 19, no. 8, pp. 2515–2527, 2018. doi: 10.1109/TITS.2017.2752202
|
[6] |
K. Nakayama, C. Zhao, L. F. Bic, M. B. Dillencourt, and J. Brouwer, " Distributed power flow loss minimization control for future grid,” International Journal of Circuit Theory and Applications, vol. 43, no. 9, pp. 1209–1225, 2015. doi: 10.1002/cta.v43.9
|
[7] |
C. D. Nicholson and W. Zhang, " Optimal network flow: a predictive analytics perspective on the fixed-charge network flow problem,” Computers &Industrial Engineering, vol. 99, pp. 260–268, 2016.
|
[8] |
T. Soares, R. J. Bessa, P. Pinson, and H. Morais, " Active distribution grid management based on robust ac optimal power flow,” IEEE Transactions on Smart Grid, vol. 9, no. 6, pp. 6229–6241, 2018. doi: 10.1109/TSG.2017.2707065
|
[9] |
J. Qin, Y. Chow, J. Yang, and R. Rajagopal, " Distributed online modified greedy algorithm for networked storage operation under uncertainty,” IEEE Transactions on Smart Grid, vol. 7, no. 2, pp. 1106–1118, 2016.
|
[10] |
K. M. Chandy, S. H. Low, U. Topcu, and H. Xu, " A simple optimal power flow model with energy storage,” in Proc. 49th IEEE Conf. on Decision and Control (CDC), pp. 1051–1057, 2010.
|
[11] |
S. Sun, J. A. Taylor, M. Dong, and B. Liang, " Distributed real-time phase balancing for power grids with energy storage,” in Proc. IEEE American Control Conf. (ACC), 2015, pp. 3032–3037.
|
[12] |
J. Lavaei and S. H. Low, " Zero duality gap in optimal power flow problem,” IEEE Transactions on Power Systems, vol. 27, no. 1, pp. 92–107, 2012. doi: 10.1109/TPWRS.2011.2160974
|
[13] |
J. F. Mota, J. M. Xavier, P. M. Aguiar, and M. Püschel, " Distributed optimization with local domains: applications in MPC and network flows,” IEEE Transactions on Automatic Control, vol. 60, no. 7, pp. 2004–2009, 2015. doi: 10.1109/TAC.2014.2365686
|
[14] |
S. Rezaei, K. Kim, and E. Bozorgzadeh, " Scalable multi-queue data transfer scheme for fpga-based multi-accelerators,” in Proc. 2018 IEEE Int. Conf. on Computer Design (ICCD), pp. 374–380.
|
[15] |
A. Billionnet and É. Soutif, " An exact method based on lagrangian decomposition for the 0-1 quadratic knapsack problem,” European Journal of Operational Research, vol. 157, no. 3, pp. 565–575, 2004. doi: 10.1016/S0377-2217(03)00244-3
|
[16] |
H. Kellerer, U. Pferschy, and D. Pisinger, " Other knapsack problems,” in Knapsack Problems, pp. 389–424, Springer, 2004.
|
[17] |
M. A. Osorio, F. Glover, and P. Hammer, " Cutting and surrogate constraint analysis for improved multidimensional knapsack solutions,” Annals of Operations Research, vol. 117, no. 1-4, pp. 71–93, 2002.
|
[18] |
M. Esmaeili and A. Mosavi, " Notice of retraction variable reduction for multi-objective optimization using data mining techniques; application to aerospace structures,” in Proc. 2nd Int. Conf. Computer Engineering and Technology (ICCET), vol. 5, pp. V5-333.
|
[19] |
G. Wu, W. Pedrycz, H. Li, D. Qiu, M. Ma, and J. Liu, " Complexity reduction in the use of evolutionary algorithms to function optimization: a variable reduction strategy,” The Scientific World Journal, vol. 2013, 2013.
|
[20] |
S. Boyd and L. Vandenberghe, Convex optimization. England, US: CUP, 2004.
|
[21] |
M. T. Heath, " Some extensions of an algorithm for sparse linear least squares problems,” SIAM Journal on Scientific and Statistical Computing, vol. 3, no. 2, pp. 223–237, 1982. doi: 10.1137/0903014
|
[22] |
A. K. Cline and I. S. Dhillon, Computation of the Singular Value Decomposition. CRC Press, 2006.
|
[23] |
M. Khorramizadeh and N. Mahdavi-Amiri, " An efficient algorithm for sparse null space basis problem using abs methods,” Numerical Algorithms, vol. 62, no. 3, pp. 469–485, 2013. doi: 10.1007/s11075-012-9599-1
|
[24] |
Q. Ba, K. Savla, and G. Como, " Distributed optimal equilibrium selection for traffic flow over networks,” in Proc. IEEE Conf. on Decision and Control, 2015.
|
[25] |
Q. Peng and S. H. Low, " Distributed optimal power flow algorithm for radial networks, Ⅰ: Balanced single phase case,” IEEE Transactions on Smart Grid, vol. 9, no. 1, pp. 111–121, 2018. doi: 10.1109/TSG.2016.2546305
|
[26] |
M. P. Abraham and A. A. Kulkarni, " ADMM-Based algorithm for solving DC-OPF in a large electricity network considering transmission losses,” IET Generation,Transmission &Distribution, vol. 12, no. 21, pp. 5811–5823, 2018.
|
[27] |
Y. Zhang, M. Hong, E. Dall’Anese, S. V. Dhople, and Z. Xu, " Distributed controllers seeking ac optimal power flow solutions using ADMM,” IEEE Transactions on Smart Grid, vol. 9, no. 5, pp. 4525–4537, 2018. doi: 10.1109/TSG.5165411
|
[28] |
J. D. Horton, " A polynomial-time algorithm to find the shortest cycle basis of a graph,” SIAM Journal on Computing, vol. 16, no. 2, pp. 358–366, 1987. doi: 10.1137/0216026
|
[29] |
R. Hariharan, T. Kavitha, and K. Mehlhorn, " Faster algorithms for minimum cycle basis in directed graphs,” SIAM Journal of Computing, vol. 38, no. 3, pp. 1430–1447, 2008.
|
[30] |
S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, " Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends ® in Machine Learning, vol. 3, no. 1, pp. 1–122, 2011.
|
[31] |
J. F. Mota, " Communication-Efficient algorithms for distributed optimization,” arXiv preprint arXiv: 1312.0263, 2013.
|
[32] |
T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms. 3th ed, MIT, 2009.
|
[33] |
R. Asadi, S. S. Kia, and A. Regan, " Cycle basis distributed ADMM solution for optimal network flow problem over bi-connected graphs,” in Proc. 54th IEEE Annual Allerton Conf. on Communication, Control, and Computing, pp. 717–723, 2016.
|
[34] |
A. Dharwadker and S. Pirzada, Graph Theory. CreateSpace Independent Publishing Platform, 2011.
|
[35] |
T. Leibfried, T. Mchedlidze, N. Meyer-Hübner, M. Nöllenburg, I. Rutter, P. Sanders, D. Wagner, and F. Wegner, " Operating power grids with few flow control buses,” in Proc. of the 6th ACM Int. Conf. on Future Energy Systems, pp. 289–294, 2015.
|
[36] |
J. Edmonds and R. M. Karp, " Theoretical improvements in algorithmic efficiency for network flow problems,” Journal of the ACM , vol. 19, no. 2, pp. 248–264, 1972. doi: 10.1145/321694.321699
|
[1] | Tai-You Chen, Wei-Neng Chen, Feng-Feng Wei, Xiao-Qi Guo, Wen-Xiang Song, Rui Zhu, Qiuzhen Lin, Jun Zhang. The Confluence of Evolutionary Computation and Multi-Agent Systems: A Survey[J]. IEEE/CAA Journal of Automatica Sinica. doi: 10.1109/JAS.2025.125246 |
[2] | Xiasheng Shi, Changyin Sun. Penalty Function-Based Distributed Primal-Dual Algorithm for Nonconvex Optimization Problem[J]. IEEE/CAA Journal of Automatica Sinica, 2025, 12(2): 394-402. doi: 10.1109/JAS.2024.124935 |
[3] | Mengli Wei, Wenwu Yu, Duxin Chen, Mingyu Kang, Guang Cheng. Privacy Distributed Constrained Optimization Over Time-Varying Unbalanced Networks and Its Application in Federated Learning[J]. IEEE/CAA Journal of Automatica Sinica, 2025, 12(2): 335-346. doi: 10.1109/JAS.2024.124869 |
[4] | Tai-You Chen, Xiao-Min Hu, Qiuzhen Lin, Wei-Neng Chen. Multi-Agent Swarm Optimization With Contribution-Based Cooperation for Distributed Multi-Target Localization and Data Association[J]. IEEE/CAA Journal of Automatica Sinica. doi: 10.1109/JAS.2025.125150 |
[5] | Zhongxin Liu, Yanmeng Zhang, Yalin Zhang, Fuyong Wang. Distributed Economic Dispatch Algorithms of Microgrids Integrating Grid-Connected and Isolated Modes[J]. IEEE/CAA Journal of Automatica Sinica, 2025, 12(1): 86-98. doi: 10.1109/JAS.2024.124695 |
[6] | Zhongyuan Zhao, Zhiqiang Yang, Luyao Jiang, Ju Yang, Quanbo Ge. Privacy Preserving Distributed Bandit Residual Feedback Online Optimization Over Time-Varying Unbalanced Graphs[J]. IEEE/CAA Journal of Automatica Sinica, 2024, 11(11): 2284-2297. doi: 10.1109/JAS.2024.124656 |
[7] | Wangli He, Yanzhen Wang. Distributed Optimal Variational GNE Seeking in Merely Monotone Games[J]. IEEE/CAA Journal of Automatica Sinica, 2024, 11(7): 1621-1630. doi: 10.1109/JAS.2024.124284 |
[8] | Feisheng Yang, Jiaming Liu, Xiaohong Guan. Distributed Fixed-Time Optimal Energy Management for Microgrids Based on a Dynamic Event-Triggered Mechanism[J]. IEEE/CAA Journal of Automatica Sinica, 2024, 11(12): 2396-2407. doi: 10.1109/JAS.2024.124686 |
[9] | Jie Hou, Xianlin Zeng, Gang Wang, Jian Sun, Jie Chen. Distributed Momentum-Based Frank-Wolfe Algorithm for Stochastic Optimization[J]. IEEE/CAA Journal of Automatica Sinica, 2023, 10(3): 685-699. doi: 10.1109/JAS.2022.105923 |
[10] | Zhe Chen, Ning Li. An Optimal Control-Based Distributed Reinforcement Learning Framework for A Class of Non-Convex Objective Functionals of the Multi-Agent Network[J]. IEEE/CAA Journal of Automatica Sinica, 2023, 10(11): 2081-2093. doi: 10.1109/JAS.2022.105992 |
[11] | Jianrui Wang, Yitian Hong, Jiali Wang, Jiapeng Xu, Yang Tang, Qing-Long Han, Jürgen Kurths. Cooperative and Competitive Multi-Agent Systems: From Optimization to Games[J]. IEEE/CAA Journal of Automatica Sinica, 2022, 9(5): 763-783. doi: 10.1109/JAS.2022.105506 |
[12] | Hossein Mirinejad, Tamer Inanc, Jacek M. Zurada. Radial Basis Function Interpolation and Galerkin Projection for Direct Trajectory Optimization and Costate Estimation[J]. IEEE/CAA Journal of Automatica Sinica, 2021, 8(8): 1380-1388. doi: 10.1109/JAS.2021.1004081 |
[13] | Xiaoxing Ren, Dewei Li, Yugeng Xi, Haibin Shao. Distributed Subgradient Algorithm for Multi-Agent Optimization With Dynamic Stepsize[J]. IEEE/CAA Journal of Automatica Sinica, 2021, 8(8): 1451-1464. doi: 10.1109/JAS.2021.1003904 |
[14] | Qing Yang, Gang Chen, Ting Wang. ADMM-based Distributed Algorithm for Economic Dispatch in Power Systems With Both Packet Drops and Communication Delays[J]. IEEE/CAA Journal of Automatica Sinica, 2020, 7(3): 842-852. doi: 10.1109/JAS.2020.1003156 |
[15] | Jonathan Tuck, David Hallac, Stephen Boyd. Distributed Majorization-Minimization for Laplacian Regularized Problems[J]. IEEE/CAA Journal of Automatica Sinica, 2019, 6(1): 45-52. doi: 10.1109/JAS.2019.1911321 |
[16] | Xiaojun Tang, Jie Chen. Direct Trajectory Optimization and Costate Estimation of Infinite-horizon Optimal Control Problems Using Collocation at the Flipped Legendre-Gauss-Radau Points[J]. IEEE/CAA Journal of Automatica Sinica, 2016, 3(2): 174-183. |
[17] | Gang Chen, Ening Feng. Distributed Secondary Control and Optimal Power Sharing in Microgrids[J]. IEEE/CAA Journal of Automatica Sinica, 2015, 2(3): 304-312. |
[18] | Zhixin Liu, Yazhou Yuan, Xinping Guan, Xinbin Li. An Approach of Distributed Joint Optimization for Cluster-based Wireless Sensor Networks[J]. IEEE/CAA Journal of Automatica Sinica, 2015, 2(3): 267-273. |
[19] | Qiming Zhao, Hao Xu, Sarangapani Jagannathan. Near Optimal Output Feedback Control of Nonlinear Discrete-time Systems Based on Reinforcement Neural Network Learning[J]. IEEE/CAA Journal of Automatica Sinica, 2014, 1(4): 372-384. |
[20] | Yanqiong Zhang, Youcheng Lou, Yiguang Hong. An Approximate Gradient Algorithm for Constrained Distributed Convex Optimization[J]. IEEE/CAA Journal of Automatica Sinica, 2014, 1(1): 61-67. |