- Research
- Open Access
A new parallel splitting augmented Lagrangian-based method for a Stackelberg game
- Xihong Yan^{1}Email author and
- Ruiping Wen^{1}
https://doi.org/10.1186/s13660-016-1047-7
© Yan and Wen 2016
- Received: 2 December 2015
- Accepted: 19 March 2016
- Published: 1 April 2016
Abstract
This paper addresses a novel solution scheme for a special class of variational inequality problems which can be applied to model a Stackelberg game with one leader and three or more followers. In the scheme, the leader makes his decision first and then the followers reveal their choices simultaneously based on the information of the leader’s strategy. Under mild conditions, we theoretically prove that the scheme can obtain an equilibrium. The proposed approach is applied to solve a simple game and a traffic problem. Numerical results about the performance of the new method are reported.
Keywords
- Stackelberg game
- variational inequality
- separable structure
- convergence
- parallel splitting method
1 Introduction
The structured system (1.1)-(1.2) can be viewed as a mathematical formulation of a one-leader-m-follower Stackelberg game where ith follower controls his decision \(x_{i}\), \(i=1,2,\ldots,m\). For the special case where the game has one leader and two followers, that is, \(m=2\), there is extensive literature on numerical algorithms [1–5]. Although the one-leader-two-follower game is helpful as a baseline model in analyzing games and their equilibrium-seeking algorithms, many application problems involve three or more followers, that is, they are formulated as the problem (1.1)-(1.2) with \(m\geq3\). However, there is less work that focuses on designing algorithms for the general case \(m\geq3\) compared to a considerable number of studies on the special case \(m=2\).
Note that the alternating direction method has been an attractive approach in that it successfully employs the Gauss-Seidel decomposition technique. However, in the method, each follower reveals his strategies sequentially, that is, the decision \(x_{i}^{k+1}\) of the follower i is revealed only when his former followers’ strategies \(x_{1}^{k+1},\ldots,x_{i-1}^{k+1}\) are available, which is not reasonable for the case where each follower is blind with the others’ strategies.
It is noted that in the schemes (1.5)-(1.6), the leader controls all decision variables which is not realistic since in many practical problems the leader only has power on his own decision variables. Based on this identified research gap, the aim of this work is to devise a new method for the problem (1.1)-(1.2), that is, a mathematical formulation of a one-leader-m-follower Stackelberg game where the leader controls λ and the ith follower controls \(x_{i}\). In the correction step of our method, only the leader’s variable λ is improved. Furthermore, we provide insights on the convergence of our method and a computational study of its performance.
The rest of the paper is organized as follows. Section 2 gives preliminaries, such as definitions and notations, which will be useful for our analysis and ease of exposition. Section 3 presents the proposed method in detail. Section 4 conducts an analysis on the global convergence of the proposed method. In Section 5, we apply the method to solve some practical problems and report the corresponding computational results. Finally, conclusions and some future research directions are stated in Section 6.
2 Preliminaries
Definition 2.1
- (a)A mapping \(f: \mathcal {R}^{n}\to\mathcal {R}^{n}\) is described as a monotone function, if$$(x_{1}-x_{2})^{T} \bigl(f(x_{1})-f(x_{2}) \bigr)\geq0,\quad \forall x_{1}, x_{2}\in \mathcal {R}^{n}. $$
- (b)A mapping \(f: \mathcal {R}^{n}\to\mathcal {R}^{n}\) is described as a strongly monotone function with modulus \(\mu>0\), if$$(x_{1}-x_{2})^{T} \bigl(f(x_{1})-f(x_{2}) \bigr)\geq\mu\|x_{1}-x_{2}\|^{2},\quad \forall x_{1}, x_{2}\in\mathcal {R}^{n}. $$
In the paper, there is a basic assumption that the mappings \(f_{i}\) (\(i=1,2,\ldots,m\)) are continuous and strongly monotone with modulus \(\mu _{f_{i}}\), respectively.
3 Parallel method
In this section, we formally state the procedure of the new parallel splitting method for solving the problem (1.1)-(1.2) and provide some insights on the method’s properties.
Algorithm 3.1
(A new augmented Lagrangian-based parallel splitting method)
- S0.
Select an initial point \(u^{0}=(x_{1}^{0},\ldots,x_{m}^{0},\lambda ^{0}) \in\mathcal{U}\), \(\epsilon>0\), \(\alpha>0\), and H, and set \(k=0\).
- S1.Simultaneously obtain solutions \(x_{i}^{k+1}\) (\(i=1,2,\ldots,m\)) by solving the following variational inequalities:respectively. Then set$$\begin{aligned} &x_{i}\in\mathcal{X}_{i},\quad \bigl(x_{i}^{\prime}-x_{i}\bigr)^{T} \biggl\{ f_{i}(x_{i})-A_{i}^{T} \biggl[ \lambda^{k}-H \biggl(\sum_{j\neq i} A_{j}x_{j}^{k}+ A_{i}x_{i}-b \biggr) \biggr] \biggr\} \ge0, \\ &\quad \forall x_{i}^{\prime}\in \mathcal{X}_{i}, \end{aligned}$$(3.1)$$ \tilde{\lambda}^{k}=\lambda^{k}- H \Biggl(\sum _{i=1}^{m} A_{i}x_{i}^{k+1}-b \Biggr). $$(3.2)
- S2.Update \(\lambda^{k+1}\) through equation (3.3),$$ \lambda^{k+1}= \lambda^{k}-\alpha\bigl( \lambda^{k}-\tilde{\lambda}^{k}\bigr). $$(3.3)
- S3.Ifstop. Otherwise, set \(k=k+1\) and go to S1.$$ \max \Bigl\{ \max_{i}\bigl\| A_{i}x_{i}^{k}-A_{i}x_{i}^{k+1} \bigr\| , \bigl\| \lambda^{k}-\lambda^{k+1}\bigr\| \Bigr\} \leq \epsilon, $$(3.4)
Remark 3.1
Remark 3.2
It is obvious that our proposed algorithm falls into the parallel splitting method since all the subproblems (3.1) can be solved in parallel by many existing efficient algorithms. Moreover, the proposed algorithm makes the best of the separable characteristic of the concerned problem (1.1)-(1.2) since only one function is involved in each subproblem. In addition, the proposed algorithm is a prediction-correction parallel splitting method. But the most significant difference from others is that only λ is corrected, which leads to less computational cost.
4 Convergence result
The convergence property of our parallel splitting algorithm is given in this section. First, we give a lemma that is useful for the convergence result.
Lemma 4.1
Proof
Based on the above analysis, the global convergence of the proposed method is presented in the following theorem.
Theorem 4.2
Proof
5 Numerical experiments
In this section, we present some numerical results by implementing our proposed algorithm in a game and a traffic problem, which demonstrate the application and performance of our proposed algorithm. All tests are performed in a MATLAB environment on a PC with Intel Core 2 Duo 3.10GHz CPU and 4GB of RAM. The section is organized as follows: First, we provide strategies for players in a game to verify the application of our algorithm. Second, we investigate the performance of the proposed algorithm by comparing it with one existing algorithm for solving a generic test problem.
Example 5.1
A simple game with one leader and three followers.
Computational results for Example 5.1
ϵ | \(\boldsymbol {u^{0}=(0,0,0,0)}\) | \(\boldsymbol {u^{0}=(1,1,1,1)}\) | Random point | |||
---|---|---|---|---|---|---|
Iter. | CPU (s) | Iter. | CPU (s) | Iter. | CPU (s) | |
10^{−4} | 30 | 0.0078 | 20 | 0.0052 | 22 | 0.0062 |
10^{−5} | 40 | 0.0090 | 28 | 0.0070 | 31 | 0.0078 |
10^{−6} | 57 | 0.0120 | 39 | 0.0090 | 42 | 0.0106 |
To further showcase the performance of the proposed algorithm, we use it to solve a generic test problem, a traffic equilibrium problem with fixed demand. Now, we introduce the problem briefly.
Example 5.2
A traffic equilibrium problem with fixed demand constraints.
Parameter setting and notations for the network
Parameter | Description | Value |
---|---|---|
a | Link connecting two nodes | |
L | Link set | |
\(f_{a}\) | Link flow on link a | |
f | Arc flow vector | |
p | Path | |
ω | O/D pairs | \(\{\omega_{1}=(1,20), \omega_{2}=(1,25), \omega _{3}=(1,24),\omega_{4}=(2,20), \omega_{5}=(3,25),\omega_{6}=(11,25)\}\) |
\(P_{\omega}\) | Set of the paths connecting O/D pair ω | |
\(d_{\omega}\) | Traffic amount between O/D pair ω | \(d_{1}=10\), \(d_{2}=20\), \(d_{3}=20\), \(d_{4}=55\), \(d_{5}=100\), \(d_{6}=30\) |
d | O/D pair traffic amount vector | |
A | Path-arc incidence matrix | A(i,j)=1 if the ith path contains link j, otherwise A(i,j)=0, \(A\in R^{55\times37}\) |
B | Path-O/D pair incidence matrix | B(i,j)=1 if the ith O/D pair contains path j, otherwise B(i,j)=0, \(B\in R^{6\times55}\) |
The link cost function \(\pmb{t_{a}(f)}\) for Example 5.2
\(t_{1}(f)=5\cdot10^{-6}f^{4}_{1}+0.5f_{1}+0.2f_{2}+50\) | \(t_{2}(f)=3\cdot10^{-6}f_{2}^{4}+0.4f_{2}+0.4f_{1}+20\) |
\(t_{3}(f)=5\cdot10^{-6}f^{4}_{3}+0.3f_{3}+0.1f_{4}+35\) | \(t_{4}(f)=3\cdot10^{-6}f_{4}^{4}+0.6f_{4}+0.3f_{5}+40\) |
\(t_{5}(f)=6\cdot10^{-6}f^{4}_{5}+0.6f_{5}+0.4f_{6}+60\) | \(t_{6}(f)=0.7f_{6}+0.3f_{7}+50\) |
\(t_{7}(f)=8\cdot10^{-6}f^{4}_{7}+0.8f_{7}+0.2f_{8}+40\) | \(t_{8}(f)=4\cdot10^{-6}f_{8}^{4}+0.5f_{8}+0.2f_{9}+65\) |
\(t_{9}(f)=10^{-6}f^{4}_{9}+0.6f_{9}+0.2f_{10}+70\) | \(t_{10}(f)=0.4f_{10}+0.1f_{12}+80\) |
\(t_{11}(f)=7\cdot10^{-6}f^{4}_{11}+0.7f_{11}+0.4f_{12}+65\) | \(t_{12}(f)=0.8f_{12}+0.2f_{13}+70\) |
\(t_{13}(f)=10^{-6}f^{4}_{13}+0.7f_{13}+0.3f_{18}+60\) | \(t_{14}(f)=0.8f_{14}+0.3f_{15}+50\) |
\(t_{15}(f)=3\cdot10^{-6}f^{4}_{15}+0.9f_{15}+0.2f_{14}+20\) | \(t_{16}(f)=0.8f_{16}+0.5f_{12}+30\) |
\(t_{17}(f)=3\cdot10^{-6}f^{4}_{17}+0.7f_{17}+0.2f_{15}+45\) | \(t_{18}(f)=0.5f_{18}+0.1f_{16}+30\) |
\(t_{19}(f)=0.8f_{19}+0.3f_{17}+60\) | \(t_{20}(f)=3\cdot 10^{-6}f_{20}^{4}+0.6f_{20}+0.1f_{21}+30\) |
\(t_{21}(f)=4\cdot10^{-6}f^{4}_{21}+0.4f_{21}+0.1f_{22}+40\) | \(t_{22}(f)=2\cdot10^{-6}f_{22}^{4}+0.6f_{22}+0.1f_{23}+50\) |
\(t_{23}(f)=3\cdot10^{-6}f^{4}_{23}+0.9f_{23}+0.2f_{24}+35\) | \(t_{24}(f)=2\cdot10^{-6}f_{24}^{4}+0.8f_{24}+0.1f_{25}+40\) |
\(t_{25}(f)=3\cdot10^{-6}f^{4}_{25}+0.9f_{25}+0.3f_{26}+45\) | \(t_{26}(f)=6\cdot10^{-6}f_{26}^{4}+0.7f_{26}+0.8f_{27}+30\) |
\(t_{27}(f)=3\cdot10^{-6}f^{4}_{27}+0.8f_{27}+0.3f_{28}+50\) | \(t_{28}(f)=3\cdot10^{-6}f_{28}^{4}+0.7f_{28}+65\) |
\(t_{29}(f)=3\cdot10^{-6}f^{4}_{29}+0.3f_{29}+0.1f_{30}+45\) | \(t_{30}(f)=4\cdot10^{-6}f_{30}^{4}+0.7f_{30}+0.2f_{31}+60\) |
\(t_{31}(f)=3\cdot10^{-6}f^{4}_{31}+0.8f_{31}+0.1f_{32}+75\) | \(t_{32}(f)=6\cdot10^{-6}f_{32}^{4}+0.8f_{32}+0.3f_{33}+65\) |
\(t_{33}(f)=4\cdot10^{-6}f^{4}_{33}+0.9f_{33}+0.2f_{31}+75\) | \(t_{34}(f)=6\cdot10^{-6}f_{34}^{4}+0.7f_{34}+0.3f_{30}+55\) |
\(t_{35}(f)=3\cdot10^{-6}f^{4}_{35}+0.8f_{35}+0.3f_{32}+60\) | \(t_{36}(f)=2\cdot10^{-6}f_{36}^{4}+0.8f_{36}+0.4f_{31}+75\) |
\(t_{37}(f)=6\cdot10^{-6}f^{4}_{37}+0.5f_{37}+0.1f_{36}+35\) |
Computational results for Example 5.2
ϵ | \(\boldsymbol {u^{0}=\operatorname{rand}(61,1)}\) | \(\boldsymbol {u^{0}=50^{*}\operatorname{ones}(61,1)}\) | \(\boldsymbol {u^{0}=100^{*}\operatorname{ones}(61,1)}\) | ||||
---|---|---|---|---|---|---|---|
Alg 3.1 | Alg in [11] | Alg 3.1 | Alg in [11] | Alg 3.1 | Alg in [11] | ||
10^{−4} | Iter. | 122 | 160 | 146 | - | 153 | - |
CPU (s) | 0.1560 | 0.1872 | 0.1092 | - | 0.1248 | - | |
10^{−5} | Iter. | 153 | 177 | 168 | - | 175 | - |
CPU (s) | 0.1716 | 0.2340 | 0.1404 | - | 0.1560 | - | |
10^{−6} | Iter. | 183 | 196 | 190 | - | 197 | - |
CPU (s) | 0.2028 | 0.2496 | 0.1872 | - | 0.2028 | - |
The numerical results from Table 4 demonstrate the preference of our algorithm over the algorithm in [11] since both the number of iterations and the CPU time of our algorithm are smaller than those of the algorithm in [11] for a random initial point and our algorithm can solve the problem while the algorithm in [11] fails to solve it for other initial points. The results verify the efficiency and effectiveness of the proposed algorithm again.
6 Conclusion
The system (1.1)-(1.2) can be considered as a mathematical formulation of a one-leader-m-follower Stackelberg game in which the leader constantly improves his strategy by determining the value of λ from strategy set \(\mathcal{R}^{l}\) while the ith follower determines his plan \(x_{i}\) from set \(\mathcal{X}_{i}\) based on the value of λ. Based on the characteristic, we design an augmented Lagrangian-based parallel splitting method to solve the system. In the method, each player can only control and improve his own decision. We establish the global convergence of the method under some suitable conditions. Finally, we conduct a computational study to demonstrate the validity and efficiency of our algorithm.
To improve the application of the proposed algorithm, we provide two research directions according to its limitations. First, the convergence of the method is proved under the condition that each player’s utility function is strongly monotone. We plan to relax the condition such that our method can be applied to more practical problems. Second, our method only serves to solve problems with a separable structure, which sounds reasonable but may not always be the case. We should improve it to solve general problems.
Declarations
Acknowledgements
This work is supported by grants from the NSF of Shanxi Province (2014011006-1).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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