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Partiallyshared pessimistic bilevel multifollower programming: concept, algorithm, and application
Journal of Inequalities and Applications volume 2016, Article number: 15 (2016)
Abstract
When multiple followers are involved in a bilevel programming problem, the leader’s decision will be affected by the reactions of these followers. For actual problems, the leader in general cannot obtain complete information from the followers so that he may be riskaverse. Then he would need a safety margin to bound the damage resulting from the undesirable selections of the followers. This situation is called a pessimistic bilevel multifollower (PBLMF) programming problem. This research considers a partiallyshared linear PBLMF programming in which there is a partiallyshared variable among the followers. The concept and solution algorithm of such a problem are developed. As an illustration, the partiallyshared linear PBLMF programming model is applied to a company making venture investments.
Introduction
Bilevel programming plays an exceedingly important role in different application fields, such as transportation, economics, ecology, engineering and others; see [1] and the references therein. It has been developed and researched by many authors; e.g., see the monographs [2–5].
When the set of solutions of the lower level problem does not reduce to a singleton, the leader can hardly optimize his choice unless he knows the follower’s reaction to his choice. In this situation, at least two approaches have been suggested: optimistic (or strong) formulation and pessimistic (or weak) formulation [3, 6, 7]. The pessimistic bilevel programming problem is very difficult [8]. As a result, most research on bilevel programming focuses on the optimistic formulation. Interested readers can refer to [1, 9] and the references therein.
This research focuses on the concept, algorithms and applications of the pessimistic bilevel programming problem. Several relative studies are reviewed for existing results of solutions and approximations results for pessimistic bilevel programming; see [10–15]. For papers discussing optimality conditions, see [16, 17]. Recently, Wiesemann et al. [8] analyzed the structural properties and presented a solvable ϵapproximation algorithm for the independent pessimistic bilevel programming problem. Based on an exact penalty function, Zheng et al. [18] proposed an algorithm for the pessimistic linear bilevel programming problem. C̆ervinka, Matonoha and Outrata [19] developed a new numerical method to compute approximate and socalled relaxed pessimistic solutions to mathematical programming with equilibrium constraints which is a generalized bilevel programming problem.
As is well known, most theoretical and algorithmic contributions to bilevel programming are limited to a specific situation with one leader and one follower. For the actual bilevel programming problems, however, the lower level problem often involves multiple decision makers. For example, in a university, the dean of a faculty is the leader, and aims to minimize the faculty annual budget. All the heads of departments in the faculty are the followers whose aims are maximizing their respective annual budget. The leader chooses an optimal strategy knowing how the followers will react. This is a typical bilevel multifollower (BLMF) programming problem. Note that the research on BLMF has been concentrated in its optimistic formulation. For example, Calvete and Galé [20] discussed the linear BLMF with independent followers and transformed such a problem into a linear bilevel problem with one leader and one follower. Lu et al. [21] generalized a framework for a special kind of BLMF, and identified nine main types of relations among followers. Lu et al. [22] considered a trilevel multifollower programming problem, and analyzed various kinds of relations between decision entities. However, some practical problems need to be modeled as a partiallyshared pessimistic BLMF programming model. Let us consider a simple example as follows.
Example 1.1
Assume that a company (i.e., leader) undertakes M projects which will be performed by M construction teams (i.e., followers), respectively. In addition to his own resources, each team usually need to use some shared resources, such as piling machines and cranes, in the company. Furthermore, assume that the construction teams are competitive, and the company cannot obtain complete information from these teams. Then the company may be riskaverse, and consequently, he protects himself against the possible worst choice of the teams. That is, his aim is to minimize the worstcase cost. Let the cost function of the corporation be \(H(x,y_{1},y_{2},\dots,y_{M},z)\). The cost function of the ith construction team is \(f_{i}(x,y_{i},z)\) and the ith team subjects constraints are \(G_{i}(x,y_{i},z)\leqslant0\) in which z is a shared resource among teams. Then the model is given as follows:
where \((y_{i},z)\) is a solution of the ith team’s problem (\(i=1,2,\dots,M\))
Note that the above problem cannot be modeled from the existing approaches. To model such a problem, the proposed study considers a partiallyshared PBLMF programming problem. The main contributions of this study are threefold: (i) the concept of a solution of the general PBLMF programming problem is presented and the related existence theorem is established; (ii) a simple algorithm based on penalty function is developed for solving a partiallyshared linear PBLMF programming problem; and (iii) we apply the proposed partiallyshared linear PBLMF programming problem to a company making venture investments.
The paper is organized as follows. In the next section, the concept of a partiallyshared PBLMF programming problem is introduced, and an equivalently penalty problem inspired from [10, 23–25] is given. In Section 3, we analyze the relationships between the original problem and its penalty problem, and then present a solution algorithm. To illustrate the feasibility and rationality of the proposed partiallyshared linear PBLMF programming model, an example of venture investments is proposed in Section 4. Finally, concluding remarks are provided in Section 5.
Concept and penalty function of partiallyshared linear PBLMF
Consider the following partiallyshared linear PBLMF programming problem in which \(M\geqslant2\) followers are involved and there is a partially shared decision variable z among followers:
where \(\Psi_{i}(x)\) is the set of solutions of the ith follower’s problem
Here, \(x,c\in\mathbb{R}^{n}\), \(y_{i},d_{i},u_{i}\in\mathbb{R}^{m_{i}}\), \(s,z,v_{i}\in\mathbb{R}^{l}\), \(A_{i}\in\mathbb{R}^{q_{i}\times n}\), \(B_{i}\in \mathbb{R}^{q_{i}\times{m_{i}}}\), \(C_{i}\in\mathbb{R}^{q_{i}\times l}\), \(b_{i}\in \mathbb{R}^{q_{i}}\), \(i=1,2,\dots,M\), X is a closed subset of \(\mathbb{R}^{n}\), and T stands for transpose.
Definition 1

(a)
Constraint region of problem (1):
$$\begin{aligned}& S= \bigl\{ (x,y_{1},y_{2},\dots,y_{M},z): x\in X, A_{i}x+B_{i}y_{i}+C_{i}z\leqslant b_{i}, y_{i},z\geqslant0, i=1,2, \dots,M \bigr\} . \end{aligned}$$ 
(b)
Projection of S onto the leader’s decision space:
$$\begin{aligned}& S(X)= \bigl\{ x\in X: \exists(y_{1},y_{2}, \dots,y_{M},z), \mbox{such that } (x,y_{1},y_{2}, \dots,y_{M},z)\in S \bigr\} . \end{aligned}$$ 
(c)
Feasible set for the ith follower \(\forall x\in S(X)\):
$$\begin{aligned}& S_{i}(x)= \bigl\{ (y_{i},z): B_{i}y_{i}+C_{i}z \leqslant b_{i}A_{i}x, y_{i},z\geqslant0 \bigr\} . \end{aligned}$$ 
(d)
The ith follower’s rational reaction set for \(x\in S(X)\):
$$\begin{aligned}& \Psi_{i}(x)= \bigl\{ (y_{i},z): (y_{i},z)\in \operatorname {Arg}\min \bigl[u_{i}^{T}y_{i}+v_{i}^{T}z: (y_{i},z)\in S_{i}(x) \bigr] \bigr\} . \end{aligned}$$ 
(e)
Inducible region or feasible region of the leader:
$$\begin{aligned}& \mathit{IR}= \bigl\{ (x,y_{1},y_{2},\dots,y_{M},z): (x,y_{1},y_{2},\dots,y_{M},z)\in S, (y_{i},z)\in\Psi_{i}(x), i=1,2,\dots,M \bigr\} . \end{aligned}$$
To introduce the concept of a solution of problem (1) (also called pessimistic solution), one usually employs the following value function \(\varphi(x)\):
Definition 2
A point \((x^{*},y_{1}^{*},y_{2}^{*},\dots,y_{M}^{*},z^{*})\in \mathit{IR}\) is called a pessimistic solution to problem (1), if
For the sake of simplicity, this study only considers a special case of \(M=2\) in problem (1), i.e.,
where \(\Psi_{i}(x)\) is the set of solutions of the ith follower’s problem
For each \(x\in S(X)\), denote by \(f(x)\) the optimal value of the following problem \(P(x)\):
Then problem (2) is equivalently transformed into the following problem P:
The dual problem of (3) is written as
Denote by \(\pi_{i}(x,y_{i},w_{i},z)=u_{i}^{T}y_{i}+v_{i}^{T}z+(b_{i}A_{i}x)^{T}w_{i}\) the ith (\(i=1,2\)) follower’s duality gap.
For \(\rho>0\), we now consider the following penalized problem \(P_{\rho}(x)\):
and denote the optimal value function by \(f_{\rho}(x)\).
The dual problem of (4) is
Furthermore, for each \(x\in S(X)\), if \(f_{\rho}(x)\) exists, then it is also the optimal value function of problem (5).
Finally, we find two penalized problems of problem P as follows.
Problem \(P_{\rho}\):
Problem \(\tilde{P}_{\rho}\):
The following section outlines the existence of solutions to problems \(P_{\rho}(x)\), \(P_{\rho}\), \(\tilde{P}_{\rho}\), and P, gives the relationships among them and presents a solution algorithm.
Algorithm of partiallyshared linear PBLMF
In order to establish theoretical results, we state the main assumption throughout the paper.
Assumption (A)
S is a nonempty compact polyhedron.
For convenience, we denote
In the sequel, denote by \(V(\mathcal{A})\) the set of vertices of \(\mathcal{A}\) for a set \(\mathcal{A}\).
The following three lemmas provide the existence of solutions to problems \(P_{\rho}(x)\), \(P_{\rho}\), and \(\tilde{P}_{\rho}\), respectively.
Lemma 3.1
Under Assumption (A), for each \(x\in S(X)\) and a fixed value of \(\rho>0\), problem \(P_{\rho}(x)\) has at least one solution in \(V(Z_{5}(x))\times V(Z_{4}^{1})\times V(Z_{4}^{2})\).
Proof
For each \(x\in S(X)\) and fixed \(\rho>0\), we have
It follows from Assumption (A) that the objective function of the linear programming problem \(P_{\rho}(x)\) is bounded from above, and hence it has at least one solution in \(V(Z_{5}(x))\times V(Z_{4}^{1})\times V(Z_{4}^{2})\). This completes the proof. □
Lemma 3.2
Under Assumption (A), for a fixed value of \(\rho>0\), problem \(P_{\rho}\) has at least one solution in \(V(Z_{2}(\rho))\times V(Z_{1}(\rho))\).
Proof
Clearly, problem \(P_{\rho}\) is a disjoint bilinear programming problem whose solution occurs at a vertex of its constraint region [26]. This completes the proof. □
Lemma 3.3
Under Assumption (A), for a fixed value of \(\rho>0\), problem \(\tilde{P}_{\rho}\) has at least one solution.
Proof
Under Assumption (A), it follows from Theorem 4.3 in [3] that \(f_{\rho}(x)\) is continuous. Hence, the result follows immediately from the Weierstrass theorem. □
For any \(\eta>0\), let \(Z_{6}:=\{(x,t): Bt\leqslant\eta(bAx)\}\) and \(Z_{7}:=\{(x,y): By\leqslant bAx\}\). To prove Theorem 3.1, we first provide the following lemma.
Lemma 3.4
For any \(\eta>0\), if \((x^{*}_{\eta},t^{*}_{\eta})\in V(Z_{6})\), then there exists \((x^{*},y^{*})\in V(Z_{7})\), such that \(x^{*}=x^{*}_{\eta}\) and \(t^{*}_{\eta}=\eta y^{*}\).
Proof
It is easy to verify that \((x^{*}_{\eta},\frac{t^{*}_{\eta}}{\eta})\in Z_{7}\). Let \((x^{1},y^{1}),\dots,(x^{r},y^{r})\) be the distinct vertices of \(Z_{7}\). Since any point in \(Z_{7}\) can be written as a convex combination of these vertices, let \((x^{*}_{\eta},\frac{t^{*}_{\eta}}{\eta})=\sum_{i=1}^{\hat{r}} \alpha^{i} (x^{i},y^{i})\), where \(\sum_{i=1}^{\hat{r}} \alpha^{i}=1\), \(\alpha^{i}>0\), \(i=1,\dots,\hat{r}\), and \(\hat{r}\leqslant r\). Then we have
Note that \((x^{i}, \eta y^{i})\in Z_{6}\). Hence, (6) implies that \(\hat{r}=1\). Because \((x^{*}_{\eta},t^{*}_{\eta})\) is a vertex of \(Z_{6}\), a contradiction results unless \(\hat{r}=1\).
Therefore, there exists a point \((x^{*},y^{*})\in V(Z_{7})\), such that \(x^{*}=x^{*}_{\eta}\) and \(t^{*}_{\eta}=\eta y^{*}\). This completes the proof. □
Next, the following result relates the solution between problems \(P_{\rho}\) and \(\tilde{P}_{\rho}\).
Theorem 3.1
Under Assumption (A), for a fixed value of \(\rho >0\), if \((x_{\rho},t_{1}^{\rho},t_{2}^{\rho},t_{3}^{\rho},t_{4}^{\rho},t_{5}^{\rho})\) is a solution of problem \(P_{\rho}\), \(x_{\rho}\) solves problem \(\tilde{P}_{\rho}\). Furthermore, \(x_{\rho}\in Q:=\{x: (x,y_{1},y_{2},z)\in V(Q^{\dagger})\}\) where
Proof
Denote the objective function of problem \(P_{\rho}\) by \(F(x,t_{1},t_{2},t_{3},t_{4},t_{5})\). Suppose that \(x_{\rho}^{*}\) solves problem \(\tilde{P}_{\rho}\). Then there exist \((t_{1}^{*},t_{2}^{*},t_{3}^{*})\in Z_{3}(\rho,x_{\rho}^{*})\) and \((t_{4}^{*},t_{5}^{*})\in Z_{1}(\rho)\), such that
Moreover, \((x_{\rho}^{*}, t_{1}^{*},t_{2}^{*},t_{3}^{*},t_{4}^{*}, t_{5}^{*})\) is a feasible point of problem \(P_{\rho}\).
Then we have
where (8) holds due to the definition of \(f_{\rho}(x_{\rho})\), (9) holds because of the optimality of \((x_{\rho},t_{1}^{\rho},t_{2}^{\rho},t_{3}^{\rho},t_{4}^{\rho},t_{5}^{\rho})\) and (10) follows from (7).
Thus, (8)(10) implies that \(x_{\rho}\) is a solution of problem \(\tilde{P}_{\rho}\).
Using the result of Lemma 3.2, we can obtain
Furthermore, by the result of Lemma 3.4, the definitions of \(Z_{2}(\rho)\) and \(Q^{\dagger}\), we find that \(x_{\rho}\in Q\). This completes the proof. □
Note that \(Q^{\dagger}\) can be referred to as the constraint region of problem (2) based on Definition 1(a).
Finally, we provide the following result which demonstrates that our penalty method is exact, and also presents the relationships between problems \(\tilde{P}_{\rho}\) and P.
Theorem 3.2
Let Assumption (A) hold, and \(\{x_{\rho}\}\) be a sequence of solutions of problem \(\tilde{P}_{\rho}\). Then there exists \(\rho^{*}>0\), such that for all \(\rho>\rho^{*}\), \(x_{\rho}\) is a solution of problem P.
Proof
Let \((y_{1}(x),y_{2}(x),z(x),w_{1}(x),w_{2}(x))\in V(Z_{5}(x))\times V(Z_{4}^{1})\times V(Z_{4}^{2})\) be a solution of problem \(P_{\rho}(x)\). For any \((\hat{y}_{1},\hat{y}_{2},\hat{z},\hat{w}_{1},\hat{w}_{2})\in V(Z_{5}(x))\times V(Z_{4}^{1})\times V(Z_{4}^{2})\), we have
In particular, choose \((\hat{y}_{i},\hat{z})\) and \(\hat{w}_{i}\) (\(i=1,2\)), such that they are solutions of problem (3) and its dual problem respectively. Then we obtain
Hence, we have
where \(\Vert \cdot \Vert _{2}\) denotes the Euclidean norm. Moreover, it follows from Assumption (A) that there exists a constant \(\delta>0\), such that
We then find that
For each \(x\in S(X)\), the number of elements of the set \(V(Z_{5}(x))\times V(Z_{4}^{1})\times V(Z_{4}^{2})\) is finite, and then there exists \(0<\rho^{*}(x)<+\infty\), such that
Since \(S(X)\) is a bounded nonempty polyhedron, there exists a constant \(\rho^{*}>0\), such that
and then \((y_{1}(x),y_{2}(x),z(x))\) is a feasible point of problem \(P(x)\). Hence, for any \(x\in S(X)\), we have
Moreover, for any \(x\in S(X)\) and \(\rho>0\), it follows from the definitions of \(f_{\rho}(x)\) and \(f(x)\) that
Therefore, for all \(\rho>\rho^{*}\), we find
where (16) and (18) follow from (15), and (17) holds because of the optimality of \(x_{\rho}\). Equations (16)(18) imply that \(x_{\rho}\) is a solution for problem P for all \(\rho>\rho^{*}\). This concludes the proof. □
Combining the results of Theorems 3.1 and 3.2, we can characterize problem (2) as a particular kind of nonlinear programming problem whose solution is related to a vertex of the constraint region.
Theorem 3.3
Under Assumption (A), there exists a solution \((x^{*} ,y_{1}^{*} ,y_{2}^{*} ,z^{*} )\) of problem (2) such that \(x^{*} \in Q\).
From the result of Theorem 3.3, we know that a solution of problem (2) may be related to a vertex of \(Q^{+}\). Hence one possible way to find the solution would be to generate all vertices of \(Q^{+}\) and test each one as a possible solution by maximizing \(P(x)\) in \((y_{1},y_{2},z)\) for fixed x. That is, solving problem (2) is equivalent to finding \(x^{*}\) with \((y^{*}_{1},z^{*}) \in\Psi_{1}(x^{*})\) and \((y^{*}_{2},z^{*}) \in\Psi_{2}(x^{*})\) such that
where for \(x_{[j]}\) (\(j=1,2,\dots,N\)) there exists a point \((x_{[j]},y_{1}^{[j]},y_{2}^{[j]},z^{[j]})\), such that \((x_{[j]},y_{1}^{[j]},y_{2}^{[j]},z^{[j]})\) are the N ordered basic feasible points for the following linear programming problem:
Rather than enumerate all vertices of the set \(Q^{+}\) explicitly, we present the following algorithm which finds a solution to problem (2).
Algorithm
 Step 0.:

Choose \(\rho>0\) and \(\gamma>1\).
 Step 1.:

Solve problem \(P_{\rho}\), and denote the solution by \((x^{\rho},t_{1}^{\rho},t_{2}^{\rho},t_{3}^{\rho},t_{4}^{\rho},t_{5}^{\rho})\).
 Step 2.:

Solve problem \(P_{\rho}(x^{\rho})\), and denote the solution by \((y_{1}^{\rho},y_{2}^{\rho},w_{1}^{\rho},w_{2}^{\rho},z^{\rho})\).
 Step 3.:

If \(\sum_{i=1}^{2} \pi_{i}(x^{\rho},y_{i}^{\rho},w_{i}^{\rho },z^{\rho})=0\), then stop, \((x^{\rho},y_{1}^{\rho},y_{2}^{\rho},z^{\rho})\) is a solution of problem (2). Otherwise, set \(\rho=\gamma\rho\) and proceed to Step 1.
An application
In this section, we apply the proposed partiallyshared PBLMF decision making model to a company making venture investments. Consider the investments with a CEO and the selected two departments of the company. All decision entities have individual objectives, constraints, and variables and do not cooperate with one another. The departments within the company are also in an uncooperative situation, but they need to use the same warehouse in this company. The CEO’s decision takes the responses of the selected departments into consideration and aims to maximize the company’s profit. At the same time, the departments fully consider the CEO’s decision, and make a rational response to maximize their own profit. Under incomplete and asymmetric information, the CEO cannot directly observe both departments’ effort and the inventory expense. Then it is difficult for the CEO to design an investment planning model that aims to maximize the company’s profit (or minimize the company’s cost). In this case, the CEO may want to create a safety margin to bound the damage resulting from undesirable selections of the two departments.
To model the above venture investment problem, the notations are introduced as follows:

(1)
The CEO (Leader)
 •:

Objective \(G_{1}\):

The aim is to maximize the company’s profit.

 •:

Variables \((x_{1},x_{2})\):
 \(x_{1}\)::

How much is used for the CEO’s investment in product 1.
 \(x_{2}\)::

How much is used for the CEO’s investment in product 2.
 •:

Constraint:
 \(H_{1}\leqslant0\)::

The total investment cost for products 1 and 2.

(2)
Two selected departments (Followers) Suppose that both department 1 (for product 1) and department 2 (for product 2) share the same warehouse.
 ⋄:

Department 1:
 •:

Objective \(G_{21}\):

The aim is to maximize his total profit which includes his own profit in product 1 and a fraction of the company’s revenue.

 •:

Variables \((y_{1},z)\):
 \(y_{1}\)::

The effort for completing product 1.
 z::

The inventory expense for product 1.
 •:

Constraints:
 \(H_{21}\leqslant0\)::

The effort and cost of department 1 that links with the CEO’s investments.
 \(H_{22}\leqslant0\)::

The maximum effort of department 1.
 \(H_{23}\leqslant0\)::

The maximum inventory expense in product 1.
 ⋄:

Department 2:
 •:

Objective \(G_{31}\):

The aim is to maximize his total profit which includes his own profit in product 2 and a fraction of the company’s revenue.

 •:

Variables \((y_{2},z)\):
 \(y_{2}\)::

The effort for completing product 2.
 z::

The inventory expense for product 2.
 •:

Constraints:
 \(H_{31}\leqslant0\)::

The effort and cost of department 2 that links with the CEO’s investments.
 \(H_{32}\leqslant0\)::

The maximum effort of department 2.
 \(H_{33}\leqslant0\)::

The maximum inventory expense in product 2.
Then a partiallyshared linear PBLMF model of the company making venture investments is given as follows:
To use the proposed algorithm, we can equivalently transform problem (19) into the following problem:
Note that both (19) and (20) have the same solutions, and their optimal values are negatives of each other.
Choose \(\rho=1\) and \(\gamma=10\). The disjoint bilinear programming problems at Step 1 are solved by a commercial optimization software package BARON [27, 28]. By using the proposed algorithm, it is easy to see that \((x_{1}^{*},x_{2}^{*},y_{1}^{*},y_{2}^{*},z^{*})^{T}=(0,1,0.5, 0, 1)^{T}\) is a solution of problem (19), and the optimal value is 9.
Conclusions
This study addresses a partiallyshared linear PBLMF programming problem in which there is a partiallyshared variable among followers. Furthermore, the study presents the concept and a solution algorithm for such a problem. Finally, the partiallyshared PBLMF model is applied to a company making venture investments. For future research, it will be interesting to propose the modified intelligent algorithms for the partiallyshared PBLMF model, to apply the model in various areas, and to explore the concept, algorithms, and applications of a PBLMF model in the referentialuncooperative situation.
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Acknowledgements
This research was supported by the National Natural Science Foundation of China (Nos. 11501233, 71471140, and 11401450).
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YZ and ZZ conceived and designed the study. YZ wrote and edited the manuscript. ZZ and LY examined all the steps of the proofs in this research and gave some advice. All authors read and approved the final manuscript.
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Zheng, Y., Zhu, Z. & Yuan, L. Partiallyshared pessimistic bilevel multifollower programming: concept, algorithm, and application. J Inequal Appl 2016, 15 (2016). https://doi.org/10.1186/s1366001509561
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DOI: https://doi.org/10.1186/s1366001509561
Keywords
 bilevel programming
 pessimistic formulation
 multiple followers