A reducibility method for the weak linear bilevel programming problems and a case study in principal-agent

: A weak linear bilevel programming (WLBP) problem often models problems involving hierarchy structure in expert and intelligent systems under the pessimistic point. In the paper, we deal with such a problem. Using the duality theory of linear programming, the WLBP problem is first equivalently transformed into a jointly constrained bilinear programming problem. Then, we show that the resolution of the jointly constrained bilinear programming problem is equivalent to the resolution of a disjoint bilinear programming problem under appropriate assumptions. This may give a possibility to solve the WLBP problem via a single-level disjoint bilinear programming problem. Furthermore, some examples illustrate the solution process and feasibility of the proposed method. Finally, the WLBP problem models a principal-agent problem under the pessimistic point that is also compared with a principal-agent problem under the optimistic point.


Introduction
Bilevel programming problems are hierarchical optimization problems in which their constraints are defined in part by another parametric optimization problem, and have been investigated by many authors. The reader can refer to the monographs and surveys [9,13,15,16,26,34,35,45,46]. instabilities.
In the Kth-Best algorithm, the advantages are that such a algorithm can obtain a global solution of WLBP problems and terminates after the finite number of iterations [49]. However, the prior condition of its implement is generating all vertices of the constraint region of WLBP problems. It is not an easy work although there are many methods proposed to obtain all vertices of a polyhedron.
In this paper, we present a reducibility method for a WLBP problem which is different from the related papers [1,47,49], and has the following features: (i) in our paper, the WLBP problem is reduced to disjoint bilinear programming problems in which the objective function and the constraint region do not involve penalty parameters.
(ii) to obtain a solution of the WLBP problem, the proposed reducibility method is finite, that is, it only requires solving two disjoint bilinear programming problems and a linear programming problem.
(iii) the proposed method is only applied to solving the WLBP problem at present. That is, it depends on the structure of the targeted problem which shows a disadvantage of the proposed method.
The contributions of the paper are as follows.
(i) In fact, the WLBP problem is a three layered structural optimization problem. An advantage of the proposed reducibility method is that the WLBP problem is reduced to a single-level scalar optimization problem (i.e. disjoint bilinear programming problem) which is easier in the model structure, theoretical analysis and algorithm design than that of the WBLP problem.
(ii) We show that the resolution of the resulting jointly constrained bilinear programming problem via transforming the WLBP problem is equivalent to the resolution of a disjoint bilinear programming problem. This may give a possibility to solve indirectly the WLBP problem via a disjoint bilinear programming problem.
(iii) Indeed, the proposed method is a method for solving the disjoint bilinear programming problem. Other methods solve such a problem but depend on the penalty parameters, for example, in [1,47]. A comparison is made with these penalty methods in Section 5.2. It is demonstrated that in most cases the proposed method can obtain the same solutions as that of penalty methods in [1,47]. However, these penalty methods lead to computational instabilities when penalty parameters are very large.
(iv) To the author's knowledge, most studies ten years ago were focused on theory analysis of weak bilevel programming problems (e.g. existence results of solutions, stability behavior and optimality condition). Not so much has been discussed about the solution algorithms. The proposed reducibility method in this paper not only enriches the research WLBP problems, but also provides a way to study the weak bilevel programming problems with special structures.
The remainder of this paper is organized as follows. In the next section, we recall the bilinear programming and WLBP problems. In Section 3, we present a duality transformation model, and reduce the WLBP problem to a disjoint bilinear programming problem in Section 4. Section 5 presents some numerical examples to illustrate the proposed method. Section 6 develops an example of principal-agent problem to describe the application of the proposed method. Finally, concluding remarks are summarized in Section 7.

Formulation and basic definitions
In this section, we first review the bilinear programming problem in detail and then give the formulation and basic definitions of the WLBP problem.

Bilinear programming problem
In general, the bilinear programming problem is formulated as follows: where , , and • stands for transpose.
In problem (1), the objective function is called bilinear if it reduces to a linear one by fixing the vector x or y to a particular value. The bilinear programming problem (1) is said to be jointly constrained bilinear programming problem [5,24,41].
A number of authors have investigated special cases of problem (1 where X and Y are given polyhedra. The variables x and y participating in the bilinear term of the objective function in problem (2) are independently constrained. Problem (2) is said to be disjoint bilinear programming problem [4,7]. It is also called separably constrained bilinear programming problem [41].
Problem (2) can be solved by the cutting plane method of Ritter [37]. Konno [27] modified this method to solve problem (2) but failed to guarantee convergence to a global solution. Very similar to that of [27], Vaish and Shetty [42] proposed a cutting plane method which has not guarantee of finite convergence. To avoid constructing expensive disjunctive facial cuts and achieve fast convergence, Ding and Al-Khayyal [18] presented two linear cutting plane method which combines the generation of polar cuts with the computation of lower bounds. Using the duality theory of linear programming, Falk [20] developed a finite branch-and-bound algorithm; Gallo and Ulkucu [21] presented a new variant of cutting plane method. Audet et al. [7] developed an exact method without any assumptions regarding boundedness of the feasible region or of the optimal objective value. Alarie et al. [4] proposed a new algorithm which combines concavity cuts and branch-and-bound for problem (2). More recently, Effati, Mansoori and Eshaghnezhad [19] applied the projection neural network to solving problem (2).
The reader interested in a more detailed overview of the different methods of problems (1) and (2) can be referred to the surveys [5,6].

WLBP problem
In this paper, we consider the following WLBP problem: where ()  x is the set of solutions to the lower level problem Here, ,   x x x y "

Duality transformation
In this section, using the duality theory of linear programming, we will transform the WLBP problem into an equivalent single-level nonconvex optimization problem. To ensure the existence of solutions of the WLBP problem, we first introduce the following two assumptions which are from (A2) The set X is a nonempty polytope.
Then, we have the following result. Proof. It follows immediately from Theorem 3.3 in [1] or Theorem 4.1 in [33].
Obviously, the dual problem of (4) is written as: , where p z  ¡ .

 x y z "
Based on the above results, we find that ,. For each X  x , the dual problem of (6) is: Since both problems (6) and (7) have the same optimal value, the WLBP problem can be equivalently transformed into a jointly constrained bilinear programming problem as follows: Clearly, the feasible region of problem (8) has a bilinear inequality which may lead to computational complexity. So, in the following, we try to avoid this bilinear term.

Reduction of (8) to a single-level disjoint bilinear programming problem
In this section, exploiting the particular form of the nonconvex optimization problem (8), we will show that the resolution of the problem (8) is equivalent to the resolution of the following disjoint bilinear programming problems (9) and (10).
Consider the following two disjoint bilinear programming problems: and µ µ , , , Before giving the relationships among problems (8)-(10), we first present some notations and lemmas. Denote the feasible regions of problems (8)-(10) respectively by Furthermore, we split  into two sets: It is easy to verify that 1 D  .
Also, we split E into two sets: The following three lemmas give the relationships among the feasible regions of problems (8)-(10).
There is a one-one correspondence between 2  and 2 E .
Proof. It is easy to verify that there is a one-one correspondence between 2  and 2 E by This completes the proof.
Proof. The result follows immediately from Lemma 4.
is also empty, which implies that D .
Thus, we have Hence,  E . This completes the proof.
The following two lemmas can guarantee the existence of solutions of problems (9) and (10) respectively. Proof. For simplicity, we denote , consider the following linear programming problem: and its dual problem is: ,.
Let 1 ()   be the marginal function defined on X by It is easy to check that 1 ()   is a concave function, and we have

VX.
Without loss of generality, suppose that * x is a solution to problem (15). Substituting x with * x in problem 13), we have Note that, the feasible region of problem (16) is not empty because of the result of Theorem 1 in [12] and the fact that the feasible region of problem (14) is not empty.
Thus, there exists a solution ** 3 to the linear programming problem (16).
Hence, problem (9)   Proof. Firstly, we introduce some notations: x v w solves problem (10), x v w is a solution to problem (8) or problem (8) is solved by a solution to problem (9).
Proof. To prove this result, we consider the following two cases: x v w is a solution to problem (10), we have Using the relationships between 2  and 2 E , i.e. (11) and (12), It is easy to check that ( ,0, , )


x v 0 is a feasible point of problem (9). Thus, D , and it follows from Lemma 4.4 that problem (9) has at least one solution. Without loss of generality, assume that * * * * ( , , , ) u x v w is a solution to problem (9). Then, we have Combining (18) and (19), ( , , , ) u   x v w , we can obtain that which implies that * * * * ( , , , ) u x v w is also a solution to problem (8).
Using the relationships between 2  and 2 E , i.e. (11) and (12), Since we cannot determine whether the set 1  is empty, we consider the following two cases: x v w is a solution to problem (8).
Furthermore, it follows from Theorem 4.1 and its proof that we can easily obtain the following result. x v w be solutions to problems (9) and (10) respectively. Then, the resolution of problem (8) is equivalent to the two disjoint bilinear programming problems (9) and (10) in the sense that 1) If 0 u   , then * * * * ( , , , ) u x v w is a solution to problem (8).
2) If 0 u   and the solution of problem (9) does not exist, then x v w is a solution to problem (8).
3) If 0 u   and problem (9) has a solution, then (22) is a solution to problem (8).
Consequently, the resolution of problem (8) is equivalent to the resolution of the disjoint bilinear programming problems (9) and (10). Therefore, the WLBP problem is reduced to a disjoint bilinear programming problem.

Experiment results
In this section, we provide two examples to illustrate solution process and feasibility of the proposed method. : ( , , , ) . y y y y  y0 • …

Solution process of Example 1
To better illustrate the proposed method in this paper, we explain definitions and theorems within the content of Example 1, and present a detailed solution process as follows.

%%
is a solution to Example 1, and the leader's optimal objective function value is -90.
Similar as above, we can obtain that : (10, 0)  x % with : (0,0,0,0)  y % is a solution of Example 2 whose optimal objective function value of the leader is -80.

Comparison results
Using penalty method and duality theory of linear programming, Aboussoror and Mansouri [1] transformed WLBP into a disjoint bilinear programming problem with a penalty parameter. Very similar to that of [1], Zheng et al. [47] presented another disjoint bilinear programming problem.
However, in our paper, the WLBP problem is reduced to two disjoint bilinear programming problems without penalty parameters. In this section, so we describe some comparison results for these disjoint bilinear programming problems.
As shown in Tables 1 and 2, the solutions of the proposed method are the same as that of the methods developed in [1,47]. So is the execution time. Unfortunately, when the penalty parameters are increased to very large (e.g. 10 9 in Examples 1 and 2), these penalty methods lead to computational instabilities. Therefore, we consider the proposed reducibility method as a feasible and an effective approach for solving WLBP problems.

A case-based example and analysis
In this section, a case study on a principal-agent (PA) problem is presented to demonstrate the applicability of the proposed model and method. Note that, it is not the first time that weak (pessimistic) bilevel optimization is applied to the PA problem. As far as known, Tsoukalas, Wiesemann and Rustem [40] first present a PA problem based on the pessimistic bilevel programming. The problem analyzed in their paper is an independent pessimistic bilevel problem.
In other words, the feasible region of the agent does not depend on the decision variables of the principal. In our paper, a class of PA problem where the feasible region of the agent depends on the decision variables of the principal is considered.

An example
In game theory, a PA problem is one in which the principal delegates a task to the agent in exchange for a wage. The PA problem can be described as follows. Suppose that the principal determines a contract x. The agent then selects an effort or an action y to perform the task for the principal. The agent selects a y to maximize his/her utility function () A y , and the principal wishes to maximize his/her utility function ( , ) P xy . , and the utility function of the principal generated by the contract x and the effort y is determined by In general, the principal cannot observe y. When designing a contract, he may be risk averse (pessimistic) or risk prone (optimistic).
With the risk prone principal, an optimistic PA (OPA) problem can be written as follows: Note that OPA problem is an optimistic bilevel programming problem [15].
With the risk averse principal, a pessimistic PA (PPA) problem can be described as follows: In fact, PPA problem is a WLBP problem discussed in this paper.

Results analysis
Many existing methods (e.g. penalty function method, Kth-Best algorithm and branch-and-bound algorithm) can solve the OPA problem. It has a global solution ( , ) (6,0,9, To further compare the performance of the OPA and the PPA methods, we compute the average and variation of the principal's objective function value for a given contract. Here the average is defined as the average between the best and worst cases, and the variation as the difference between the best and worst cases of the principal's objective function values. Clearly, the average value of the PPA method (i.e. 52.5) is greater than that of the OPA method (i.e. 43.5). generates the agent's rational reaction set whose average of the principal's objective function value is greater than that by (0,6)  x and with a smaller variation. In other words, a contract designed by the risk averse principal is superior to that by the risk prone principal at least for this example.
As shown in the above results analysis, using the pessimistic (weak) bilevel programming to model the PA problem is of practical significance. Therefore, researches on how to solve weak bilevel programming problem is still attractive.

Conclusions and further study
In this paper, we consider a weak linear bilevel programming (WLBP) problem which is NP-hard. Using the duality theory of linear programming, the WLBP problem is equivalently transformed into a jointly constrained bilinear programming problem which is then reduced to a disjoint bilinear programming problem. Finally, in Theorem 4.2, we have shown that the resolution of this jointly constrained bilinear programming problem is equivalent to the resolution of a disjoint bilinear programming problem. Therefore, this may provide a possibility to solve the WLBP problem via a disjoint bilinear programming problem. For future research, it is interesting to discuss the weak nonlinear bilevel programming problems.