 Research
 Open Access
 Published:
A new parallel splitting augmented Lagrangianbased method for a Stackelberg game
Journal of Inequalities and Applications volume 2016, Article number: 108 (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.
Introduction
In our work, we concentrate on typical structured variational inequality problems which are mathematically described as follows: Find a point \(w^{*}\in\Omega\) such that
where
\(\mathcal{X}_{i}\subseteq\mathcal{R}^{n_{i}}\), and \(f_{i}: \mathcal {X}_{i}\to\mathcal{R}^{n_{i}} \) (\(i=1,2,\ldots,m\)) are mappings. In the work, we assume that \(\mathcal{X}_{i}\) (\(i=1,2,\ldots,m\)) are nonempty, closed, and convex sets, the mappings \(f_{i}\) (\(i=1,2,\ldots,m\)) are monotone, and the solution of the problem exists. By attaching a Lagrange multiplier vector \(\lambda\in R^{l}\) to the linear constraint \(\sum_{i=1}^{m} A_{i}x_{i}=b\), the above problem can be converted to the following form: Find a point \(u^{*}\in\mathcal {U}=\prod_{i=1}^{m}\mathcal{X}_{i}\times R^{l}\) such that
where
The structured system (1.1)(1.2) can be viewed as a mathematical formulation of a oneleadermfollower 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 oneleadertwofollower game is helpful as a baseline model in analyzing games and their equilibriumseeking 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 \(m=2\).
For the general case of the problem (1.1)(1.2), we can directly employ the classical augmented Lagrangian method proposed by [6]. However, the classical augmented Lagrangian method may prevent us from enjoying the separable structure of problem (1.1)(1.2), since the subproblems obtained in the steps of the method involve coupled variables. Frequently used and powerful ideas for dealing with the difficulty are decomposition techniques. Based on the techniques, there are two most noteworthy categories of decomposition methods for the problem (1.1)(1.2) in the literature: the alternating direction method and the parallel splitting augmented Lagrangian method. The alternating direction method has obtained recognition as a benchmark method to solve the problem (1.1)(1.2), ever since it was proposed by Gabay and Mercier [7]. The application of the solution method has been extended in the past decade to cover a variety of areas [8–10]. The basic iterative scheme of the method for solving the problem (1.1)(1.2) is as follows: When a decision \(u^{k}=(x_{1}^{k},x_{2}^{k},\ldots,x_{m}^{k}, \lambda^{k} )\) is provided, the method gets \(x_{1}^{k+1},x_{2}^{k+1},\ldots,x_{m}^{k+1}\) by solving the following system:
Then update λ by equation (1.4),
where H is a symmetric positive definite matrix.
Note that the alternating direction method has been an attractive approach in that it successfully employs the GaussSeidel 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 \(x_{1}^{k+1},\ldots,x_{i1}^{k+1}\) are available, which is not reasonable for the case where each follower is blind with the others’ strategies.
As we mentioned, the other efficient method to deal with the problem (1.1)(1.2) is socalled the parallel splitting augmented Lagrangian method proposed by He [11] based on Jacobian decomposition. Compared to the alternating direction method, the speciality of the parallel splitting augmented Lagrangian method is that it simultaneously solves the following subproblems to obtain \(x_{i}^{k+1}\) (\(i=1,2,\ldots,m\)):
which means that the followers make their choices simultaneously. However, there are few studies to address the convergence of the scheme (1.5)(1.4), which results in improved methods where the output provided by (1.5) is corrected by a further correction step [12–15]. All these methods are designed by adding a correction step as follows:
where \(\alpha_{k}\) is a stepsize, \(\tilde{u}^{k}\) is output of (1.5) which is called a predictor, and \(d(u^{k}\tilde{u}^{k})\) is a descent direction at \(u^{k}\). The schemes may be understood in the context of a game in the following way: When the leader provides a decision \(u^{k}=(x_{1}^{k},\ldots,x_{m}^{k}, \lambda^{k} )\), all followers decide their strategies \(\tilde{x}_{i}^{k}\) (\(i=1,2,\ldots,m\)) simultaneously by solving the corresponding subproblems in (1.5), respectively. Then, based on the feedback \(\tilde{x}_{i}^{k}\) (\(i=1,2,\ldots,m\)) from the followers, the leader improves his strategy by (1.6).
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 oneleadermfollower 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.
Preliminaries
For the convenience of the analysis in the paper, this section provides some basic definitions and notations. An ndimensional Euclidean space is denoted by \(\mathcal {R}^{n}\). All vectors used in the paper mean column vectors. For ease of exposition, we use the vector \((x_{1},\ldots,x_{m})\) to represent \((x_{1}^{T},\ldots,x_{m}^{T})^{T}\), where T represents the transpose operator. \(\delta_{\max}(A)\) denotes the largest eigenvalue of square matrix A. For any symmetric and positive definite matrix G, we denote by \(\x\_{G}:=\sqrt{x^{T}Gx}\) its Gnorm. In the work, we define
such that \(\uu^{*}\^{2}_{G}:=\A_{1}x_{1}A_{1}x_{1}^{*}\_{\alpha H}^{2}+\cdots+\A_{m}x_{m}A_{m}x_{m}^{*}\_{\alpha H}^{2}+\\lambda\lambda ^{*}\_{H^{1}}^{2}\).
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.
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 Lagrangianbased 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:
$$\begin{aligned} &x_{i}\in\mathcal{X}_{i},\quad \bigl(x_{i}^{\prime}x_{i}\bigr)^{T} \biggl\{ f_{i}(x_{i})A_{i}^{T} \biggl[ \lambda^{k}H \biggl(\sum_{j\neq i} A_{j}x_{j}^{k}+ A_{i}x_{i}b \biggr) \biggr] \biggr\} \ge0, \\ &\quad \forall x_{i}^{\prime}\in \mathcal{X}_{i}, \end{aligned}$$(3.1)respectively. Then set
$$ \tilde{\lambda}^{k}=\lambda^{k} H \Biggl(\sum _{i=1}^{m} A_{i}x_{i}^{k+1}b \Biggr). $$(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.
If
$$ \max \Bigl\{ \max_{i}\bigl\ A_{i}x_{i}^{k}A_{i}x_{i}^{k+1} \bigr\ , \bigl\ \lambda^{k}\lambda^{k+1}\bigr\ \Bigr\} \leq \epsilon, $$(3.4)stop. Otherwise, set \(k=k+1\) and go to S1.
Remark 3.1
Now, we conduct some analysis on the proposed algorithm. From (3.2) and (3.3), we can deduce that \(\sum_{i=1}^{m} A_{i}x_{i}^{k+1}=b\) if \(\lambda^{k+1}=\lambda^{k}\). Then we have \(\sum_{j\neq i} A_{j}x_{j}^{k}+ A_{i}x_{i}^{k+1}b=0\) under the condition that \(A_{i}x_{i}^{k+1}=A_{i}x_{i}^{k}\) (\(i=1,2,\ldots,m\)). Furthermore, according to (3.1), there exist the following inequalities and equation:
and
Thus, it is obvious that \((x_{1}^{k+1},\ldots,x_{m}^{k+1},\lambda^{k+1})\in \mathcal{U}\) is a solution of the oneleadermfollower game. Based on the analysis, we conclude that for a given small enough ϵ, the proposed method with termination condition (3.4) can obtain an approximation solution \((x_{1}^{k+1},\ldots,x_{m}^{k+1},\lambda ^{k+1})\in\mathcal{U}\) for the concerned game, that is, the stopping criterion (3.4) in the method is reasonable.
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 predictioncorrection parallel splitting method. But the most significant difference from others is that only λ is corrected, which leads to less computational cost.
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
Given \(\lambda^{k}\) by the leader and \(x_{i}^{k}\) by the ith follower \((i=1,2,\ldots,m)\) at iteration k, the strategy \((x_{1}^{k+1},\ldots,x_{m}^{k+1},\lambda^{k+1})\) in the next iteration satisfies
Proof
We assume that there exists an optimal solution \(u^{*}=(x_{1}^{*},\ldots,x_{m}^{*},\lambda^{*})\in\mathcal{U}\). Using (1.1) with \(u=u^{k+1}=(x_{1}^{k+1},\ldots,x_{m}^{k+1},\lambda^{k+1})\in\mathcal {U}\), we have
On the other hand, let \(x_{i}^{\prime}=x_{i}^{*} \) (\(i=1,2,\ldots,m\)) in each inequality of (3.1). It is easy to obtain
From the summation of all the inequalities included in (4.1) and (4.2) and the equality \(\sum_{i=1}^{m} A_{i}x_{i}^{*}=b\), we have
Rearranging the above inequality and taking account of the strong monotonicity of all the functions \(f_{i}\) (\(i=1,2,\ldots,m\)) and \(\sum_{i=1}^{m} A_{i}x_{i}^{*}=b\), we obtain
Note that b can be replaced by \(\sum_{i=1}^{m} A_{i}x_{i}^{*}\). Then
Thus,
Now, we focus on the terms \(\A_{i}x_{i}^{k+1}A_{i}x_{i}^{*}\^{2}_{\alpha H} \) (\(i=1,2,\ldots,m\)).
Since
it follows that
Adding all formulas in (4.4) and (4.5), we get the following inequality:
Since
from (4.6), we get
The proof is completed. □
Lemma 4.1 indicates that
where G is defined by (2.1).
Based on the above analysis, the global convergence of the proposed method is presented in the following theorem.
Theorem 4.2
Let m be the number of followers. Suppose that for each \(i\in\{ 1,2,\ldots,m\}\), \(f_{i}(x_{i})\) is continuous and strongly monotone on \(\mathcal {X}_{i}\subseteq\mathcal {R}^{n_{i}}\). Moreover, if
and
the sequence \(\{u^{k}\}\) generated by the proposed method converges to an optimal solution of the problem (1.1)(1.2).
Proof
From Lemma 4.1 and (4.8), we have
The two terms on the right side of the above inequality are negative due to the conditions of the theorem. Thus,
which means that \(\{u^{k}\}\) is a bounded sequence generated by the developed method. Consequently,
which means that
Thus,
On the other hand, for each \(i\in\{1,2,\ldots,m\}\), \(x_{i}^{k+1}\) satisfies the following inequality:
Moreover, there exist cluster points for the sequence \(\{\lambda^{k}\}\) due to the boundedness of \(\{\lambda^{k}\}\) implied by the boundedness of \(\{u^{k}\}\). Let \(\lambda^{*}\) be one of cluster points such that it is a limit of a convergent subsequence \(\{\lambda^{k_{j}}\}\). From the limit of (4.12) along this subsequence, it follows that
From (4.13) and (4.11), we can assert that the sequence generated by the proposed method is globally convergent. This completes the proof. □
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.
We begin the computational study by solving a oneleaderthreefollower game where each follower i decides a pure strategy \(s_{i}\) (\(i=1,2,3\)). The problem is given by the following programs,
The game is a revised version of a game considered by [16–18]. We solve the game using the proposed algorithm to obtain a solution which matches with the analytic optimal solution, that is, \(s_{1}=1\), \(s_{2}=0\), \(s_{3}=1\). Furthermore, we investigate the effect of initial points and optimality tolerances on the performance of our algorithm. In the tests, initial points are \(u_{0}=(0,0,0,0)\), \(u_{0}=(1,1,1,1)\), or a random point in \((0,1)^{4}\). Optimality tolerances are \(\epsilon=10^{4}\), \(\epsilon=10^{5}\) or \(\epsilon=10^{6}\). In our experiments, the parameter setting is \(\alpha =0.8\) and \(H=0.9I\). The associated numerical results are recorded in Table 1 where Iter. means the number of iterations and CPU means CPU time. The numerical results from Table 1 confirm the validity and efficiency of our method. Moreover, we observe that the proposed algorithm is robust to the initial points.
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.
The problem is always selected as a test case; for example, see [13, 19, 20]. Its network is shown in Figure 1 where there are 25 nodes, 37 links, and 55 paths. Other parameters and notations are summarized in Table 2.
We define variables \(x_{p}\) as the traffic flow of path p. Then arc flow vector f is calculated by the following formula:
Moreover, based on the link travel cost vector denoted by \(t(f)=\{ t_{a}, a \in L\}\) whose expressions are given in Table 3, the travel cost vector θ can be formulated as follows:
Hence, the problem is converted to a variational inequality as
where \(S=\{x\in\mathcal{R}^{55}Bx=d,x\geq0\}\).
We implement our algorithm for this problem. First, the decision variable vector x is partitioned into three parts,
where \(x_{1}\in\mathcal{R}^{25}\), \(x_{2}\in\mathcal{R}^{15}\), and \(x_{3}\in\mathcal{R}^{15}\). Subsequently, matrices A, B, and θ are partitioned, respectively, as follows:
where \(A_{1}\in\mathcal{R}^{25\times37}\), \(A_{2}\in\mathcal {R}^{15\times37}\), \(A_{3}\in\mathcal{R}^{15\times37}\), \(B_{1}\in\mathcal {R}^{6\times25}\), \(B_{2}\in\mathcal{R}^{6\times15}\), \(B_{3}\in\mathcal {R}^{6\times15}\), and θ is partitioned in the same way as x.
Based on the above partitions, the resulting traffic problem is as follows:
and
where \(S=\{(x_{1},x_{2}, x_{3})B_{1}x_{1}+B_{2}x_{2}+B_{3}x_{3}=d, x_{1}\geq0, x_{2}\geq0, x_{3}\geq0 \}\).
In order to show the efficiency and effectiveness of our algorithm, we conduct numerical experiments on the performance of the proposed algorithm and the parallel splitting augmented Lagrangian method in [11] since both methods are used for a oneleaderthreefollower game. The performance of the two algorithms with different initial points (\(u^{0}=\operatorname{rand}(61,1)\), \(u^{0}=50^{*}\operatorname{ones}(61,1)\), and \(u^{0}=100^{*}\operatorname{ones}(61,1)\)) and optimality tolerance (\(\epsilon=10^{4}\), \(\epsilon=10^{5}\), and \(\epsilon=10^{6}\)) for the traffic equilibrium problem with fixed demand constraints (Example 5.2) is stated in Table 4. Here, Alg means algorithm and ’’ means failure. In these tests, the common parameters of the two methods are the same, that is, \(H=\beta I\), where \(\beta=0.8\) and I is the identity matrix, and the maximum number of iterations is 5,000.
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.
Conclusion
The system (1.1)(1.2) can be considered as a mathematical formulation of a oneleadermfollower 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 Lagrangianbased 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.
References
 1.
Eckstein, J, Bertsekas, DP: On the DouglasRachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293318 (1992)
 2.
Fukushima, M: Application of the alternating direction method of multipliers to separable convex programming problems. Comput. Optim. Appl. 1, 93111 (1992)
 3.
Chen, G, Teboulle, M: A proximalbased decomposition method for convex minimization problems. Math. Program. 64, 81101 (1994)
 4.
Tseng, P: Alternating projectionproximal methods for convex programming and variational inequalities. SIAM J. Optim. 7, 951965 (1997)
 5.
Han, D, He, H, Yang, H, Yuan, X: A customized DouglasRachford splitting algorithm for separable convex minimization with linear constraints. Numer. Math. 127, 167200 (2014)
 6.
Hestenes, MR: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303320 (1969)
 7.
Gabay, D, Mercier, B: A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl. 2, 1740 (1976)
 8.
Esser, E: Applications of Lagrangianbased alternating direction methods and connections to split Bregman. CAM Report 0931, UCLA (2009)
 9.
Lin, Z, Chen, M, Ma, Y: The augmented lagrange multiplier method for exact recovery of corrupted lowrank matrices. UILUENG 092215, UIUC (2009)
 10.
Yang, JF, Zhang, Y: Alternating direction algorithms for \(\ell_{1}\)problems in compressive sensing. SIAM J. Sci. Comput. 33, 250278 (2011)
 11.
He, BS: Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities. Comput. Math. Appl. 42, 195212 (2009)
 12.
Tao, M: Some parallel splitting methods for separable convex programming with \(O (1/t)\) convergence rate. Pac. J. Optim. 10, 359384 (2014)
 13.
Wang, K, Xu, L, Han, D: A new parallel splitting descent method for structured variational inequalities. J. Ind. Manag. Optim. 10, 461476 (2014)
 14.
Jiang, ZK, Yuan, XM: New parallel descentlike method for solving a class of variational inequalities. J. Optim. Theory Appl. 145, 311323 (2010)
 15.
Han, D, Yuan, X, Zhang, W: An augmented Lagrangian based parallel splitting method for separable convex minimization with applications to image processing. Math. Comput. 83, 22632291 (2014)
 16.
Facchinei, F, Fischer, A, Piccialli, V: Generalized Nash equilibrium problems and Newton methods. Math. Program. 117, 163194 (2009)
 17.
Facchinei, F, Kanzow, C: Penalty methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 20, 22282253 (2010)
 18.
Han, D, Zhang, H, Qian, G, Xu, L: An improved twostep method for solving generalized Nash equilibrium problems. Eur. J. Oper. Res. 216, 613623 (2012)
 19.
Nagurney, A, Zhang, D: Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Academic, Boston (1996)
 20.
He, BS, Xu, Y, Yuan, XM: A logarithmicquadratic proximal predictioncorrection method for structured monotone variational inequalities. Comput. Optim. Appl. 35, 1946 (2006)
Acknowledgements
This work is supported by grants from the NSF of Shanxi Province (20140110061).
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript.
Rights and permissions
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.
About this article
Cite this article
Yan, X., Wen, R. A new parallel splitting augmented Lagrangianbased method for a Stackelberg game. J Inequal Appl 2016, 108 (2016). https://doi.org/10.1186/s1366001610477
Received:
Accepted:
Published:
Keywords
 Stackelberg game
 variational inequality
 separable structure
 convergence
 parallel splitting method