New partial cooperation model for bilevel programming ... - IEEE Xplore

Journal of Systems Engineering and Electronics Vol. 22, No. 2, April 2011, pp.263–266 Available online at www.jseepub.com

New partial cooperation model for bilevel programming problems Shihui Jia1,2,3 , Zhongping Wan1,* , Yuqiang Feng2 , and Guangmin Wang2 1. School of Mathematics and Statistic, Wuhan University, Wuhan 430072, P. R. China; 2. Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan University of Science and Technology, Wuhan 430081, P. R. China; 3. Mathematics Department, College of Sciences, Wuhan University of Science and Technology, Wuhan 430081, P. R. China

inf F (x, y)

Abstract: Partial cooperation models are studied for many years to solve the bilevel programming problems where the follower’s optimal reaction is not unique. However, in these existed models, the follower’s cooperation level does not depend on the leader’s decision. A new model is proposed to solve this deficiency. It is proved the feasibility of the new model when the reaction set of the lower level is lower semi-continuous. And the numerical results show that the new model has optimal solutions when the reaction set of the lower level is discrete, lower semi-continuous and nonlower semi-continuous.

Keywords: bilevel programming, reaction set, optimistic model, pessimistic model, partial cooperation model, cooperation level. DOI: 10.3969/j.issn.1004-4132.2011.02.012

1. Introduction A bilevel programming (BLP) problem can be viewed as a static version of the noncooperative two-player game introduced by [1] in the context of unbalanced economic markets. Generally speaking, the decision vector is partitioned among the players, the upper level decision maker (leader) controls one part, say x, and the lower level decision maker (follower) controls the other part, say y. Not only the leader’s decision-making can be able to influence the behavior of the follower, but also it may be simultaneously affected by the follower’s action. In other words, each player wants to optimize its own objective, but is affected by the action of the other players. Hence, perfect information is assumed. In this paper, we consider the following BLP problem [2] Manuscript received September 13, 2009. *Corresponding author. This work was supported by the National Natural Science Foundation of China (70771080), the National Science Foundation of Hubei Province (20091107), and Hubei Province Key Laboratory of Systems Science in Metallurgical Process (B201003).

(1)

x∈X

where y solves inf f (x, y) y

gi (x, y)  0, i = 1, 2, . . . , m

s.t.

Define the set Y by Y := {y |gi (x, y)  0, i = 1, 2, . . . , m} When M (x) is a singleton y˜(x) for any x ∈ X, a strategy x¯ ∈ X is called a Stackelberg solution [3 – 7] for the leader if inf F (x, y(x)) = F (¯ x, y(¯ x)) (2) x∈X

When M (x) is not a singleton for at lease one x ∈ X, the behavior of the leader depends on the choice of the follower. Two extreme possibilities have been already considered. The first one, called “optimistic” (full-cooperation), assumes that the follower will choose the one that favors the leader’s interest most, which can be written as [8] Find x¯ ∈ X such that inf

inf

x∈X y∈M(x)

F (x, y) =

inf

y∈M(¯ x)

F (¯ x, y)

(3)

Let S s be the set of solutions to the above optimistic problem and v s be the optimal value. The second possibility, called “pessimistic” (noncooperation) [8, 9], assums that the follower will select the worst choice for the leader, which has the following mathematical model Find x¯ ∈ X such that inf

sup F (x, y) = sup F (¯ x, y)

x∈X y∈M(x)

y∈M(¯ x)

(4)

Let S ω be the set of solutions to this pessimistic problem and v ω be the optimal value.

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Between the “optimistic” and “pessimistic”, it is plausible that the follower’s cooperation can be partial, which is called partial cooperation model [10]. Define β as the degree of cooperation from the follower, where 0  β  1 (β = 1 represents a fully cooperative relationship and β = 0 represents a fully non-cooperative relationship). Then, the problem is described Find x ∈ X such that inf [β

x∈X

inf

y∈M(x)

Find y¯ ∈ Y such that inf f (x, y) = f (x, y¯)

y∈Y

Let us consider β(x) = (1 − x)/2 with x ∈ [−1, 1]. We have S s = {−1}, v s = −2 S ω = {−1/2}, v ω = −1/4 √ √ p p = −1/ 3, vD = −2 3/9 SD

F (x, y) +

(1 − β) sup F (x, y)] = F (¯ x, y)

(5)

y∈M(x)

i

Let S be the set of solutions to this partial cooperation model and v i be the optimal value. Although these existed models have been successfully applied in different domains and there have been nearly dozens of algorithms proposed for solving them, the follower’s cooperation level still does not depend on the leader’s choice.

p  vω . We observe :v s  vD Remark 1 In the case M (x) is a countable set, that is M (x) = {(yn (x), n ∈ N }, (7) can also be approved.

3. Lower semi-continuous reaction set Suppose that M (x) is not discrete for at least one x ∈ X and the solutions to the following (8) and (9) exist. inf

F (x, y)

(8)

sup F (x, y)

(9)

y∈M(x)

2. An example of discrete reaction set

y∈M(x)

Suppose that M (x) is discrete and has a finite number of elements. More precisely, let

p We still consider model (6). Let SC be the solutions to p be the optimal value. Again we have model (6) and vC

M (x) := {y j (x), j = 1, 2, . . . , m(x)}

p v s  vC  vω

Incorporating β(x), the follower’s degree of cooperation, model (5) becomes the following optimization model inf [β(x)

x∈X

inf

y∈M(x)

F (x, y) +

(1 − β(x)) sup F (x, y)]

X = Y = [−1, 1] (6)

y∈M(x)

where β(x) is a random function which only needs to satisfy 0  β(x)  1 for any x ∈ X. p p Let SD be the set of solutions to the model (6) and vD be the optimal value. For any random function β(x), it is proved that p  vω (7) v s  vD Then, we can obtain a lower value than the pessimistic value. An example satisfying the above condition is given as follows. Example 1 [11] Let X = Y = [−1, 1] F (x, y) = x + y,

Consider the following example. Example 2 [11] Let

  f (x, y) = y 2 − x4 

In this case M (x) = {x2 , −x2 } is the solution set to the following problem

F (x, = xy  y) 

 x + 1/2 2   1 Then, for any x ∈ X, M (x) = x − , 0 is the set of 2 solutions to the minimum model (1). Let  x, 0  x  1 β(x) = −x, −1  x < 0 f (x, y) = max 0, y y −

we have

S s = {−1},

v s = −1

S ω = {[0, 1]}, p = 1/2, SC

vω = 0

p vC = −1/8

p  vω . In this case we still have v s  vC Theorem 1 Let X, Y be compact. Assume that the following assumptions are satisfied. (i) F (x, y) and f (x, y) are lower semi-continuous functions on X × Y .

Shihui Jia et al.: New partial cooperation model for bilevel programming problems

(ii) Any (x, y) ∈ X × Y and for any {xn } converging to x, there exists a sequence {yn } such that lim sup f (xn , yn )  f (x, y) n→∞

(iii) β(x) is a continuous function on X. (iv) M (x) is a lower semi-continuous multivalued function [12] on X. Then there exists at least a solution to the partial cooperation model (6) with index β(x). Proof Since f is lower semi-continuous, M (x) is a closed graph multivalued function, i.e., for any x ∈ X and for any {xn } conveging to x, lim sup M (xn ) ⊆ M (x),

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4. Non-lower semi-continuous reaction set When M (x) is not a lower semi-continuous multivalued function, the set M (x, ε) := {y ∈ Y : f (x, y)  ε + inf f (x, y)} y∈Y

can be considered. In [13], it is shown that under suitable assumptions, the function M (x, ε) is continuous on X for any ε > 0. Thus, we define a regularization for this new partial cooperation model in the following way inf [β(x, ε) x

inf

y∈M(x,ε)

F (x, y) +

n

i.e., for any y0 ∈ Y and for any {yn } converging to y0 with ynk ∈ M (xnk ) for a selection (nk ), we have y0 ∈ M (x). In fact, for any y satisfying lim ynk = y, by using assumptions (i) and (ii), we have

k→∞

f (x, y)  lim inf inf f (xnk , y) 

(1 − β(x, ε))

k→∞

F (x, y) = −xy,

lim sup inf f (xnk , y)  inf f (x, y) y∈Y

y∈Y

By using assumption (iv), for any x ∈ X and for any {xn } converging to x, we have k

lim M (xn ) = M (x)

n→∞

Then, by using assumptions (i) and (iii), for any x ∈ X and for any {xn } converging to x, we have inf

F (x, y) + (1 − β(x)) sup F (x, y) =

lim {β(xn )

(1 − β(xn )) lim {β(xn )

n→∞

(1 − β(xn ))

inf

F (x, y) +

sup

F (x, y)} 

y∈M(xn )

y∈M(xn )

inf

lim inf F (xn , y) +

sup

lim inf F (xn , y)} =

y∈M(xn ) n→∞

y∈M(xn ) n→∞

lim inf {β(xn ) n→∞

(1 − β(xn ))

inf

F (xn , y) +

sup

F (xn , y)}

y∈M(xn ) y∈M(xn )

S ω = φ,

Hence, the function β(x)

inf

y∈M(x)

Sεp

F (x, y) + (1 − β(x)) sup F (x, y)

is lower semi-continuous on X.

  ⎧ 1 ⎪ ⎪ {1}, x ∈ 0, ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨ 1 M (x) = [0, 1], x = 2 ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 1 ⎪ ⎩{0}, x ∈ ,1 2 is the set of solution to the minimum problem f (x, y). Because M (x) is not lower semi-continuous at x = 1/2, by using model (4) or (6), the solutions do not exist. But, by using model (10), we have ⎧    1 − 2ε 1 ⎪ ⎪ ⎪ , 1 , x ∈ 0, − ε ⎪ ⎪ 1 − 2x 2 ⎪ ⎪ ⎪ ⎪   ⎨ 1 1 M (x, ε) = [0, 1], x ∈ − ε, + ε ⎪ 2 2 ⎪ ⎪ ⎪     ⎪ ⎪ ⎪ 2ε 1 ⎪ ⎪ + ε, 1 , x∈ ⎩ 0, 2x − 1 2

y∈M(x)

n→∞

  1 f (x, y) = y x − 2

Then, for any x ∈ X

Hence, f (x, y) = inf f (x, y), i.e. y ∈ M (x).

y∈M(x)

(10)

X = Y = [0, 1]

lim inf f (xnk , ynk ) =

β(x)

F (x, y)]

Let Sεp be the solutions to problem (10) and vεp be the optimal value. In this case, we still have v s  vεp  v ω . Example 3 [11] Let

k→∞ y∈Y

k→∞ y∈Y

sup

y∈M(x,ε)

y∈M(x)



vω = −

1 2

√ ε 1 = −√ 2 2 + 4ε

1 vεp = − − 2ε, 2

ε>0

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5. Conclusion The existed models can not assure the follower’s cooperation level depends on the leader’s choice, this paper proposes model (6) to solve this deficiency. Theories and examples show that the optimal value is feasible. In future study, we will use the first and second order optimal conditions and the determination of the structure of the optimal solution set [14] to transfer model (6) to a single-level model. The algorithms will also be included for future investigation.

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and applications. Plenum Press, 1996: 271–282. [12] M. B. Lignola, J. Morgan. Semi continuity of marginal functions in a sequential setting. Optimization, 1992, 24: 241–253. [13] M. B. Lignola, J. Morgan. Topological existence and stability for Stackelberg problems. Journal of Optimization Theory and Applications, 1995, 84(1): 145–169. [14] S. J. Xue. Determining the optimal solution set for linear fractional programming. Journal of Systems Engineering and Electronics, 2002, 3(3): 40–45.

Biographies Shihui Jia was born in 1979. She is a Ph.D. in Wuhan University and a lecturer in Wuhan University of Science and Technology. Her main research interests include theory and method of optimization and system engnineering. E-mail: [email protected]

Zhongping Wan was born in 1959. He is a professor and Ph.D. supervisor in Wuhan University. His main research interests include optimization theory, algorithms and applications in electricity markets. E-mail: [email protected]

Yuqiang Feng was born in 1975. He is an associate professor and master supervisor in Wuhan University of Science and Technology. His main research interests include nonlinear analysis, differential equations and so on. E-mail:[email protected]

Guangmin Wang was born in 1978. He is a lecturer in China University of Geosciences. His research interests focus on theory and methods of optimization and system engineering. E-mail: [email protected]