Chapter 2 Literatures Review
2.3 The Variables of the Bilevel Programming
2.3.1 Continuous Variables
The hierarchical structure of the BLP problem imposes a strict order on the selection of the decision variables each planner controls. That is, the follower decision level executes its policies after, and in view of, the decision of the leader level, and the leader level optimizes its objective independently over the reactions from the follower level [39, 40]. (Yi-Hsin Liu, Stephen M. Hart and Thomas H. Spencer, 1994-1995)
Let the vectors a,c,xRn1, b,d,yRn2, and uRm. Further, let A and B be two matrices with size m×n1 and m×n2, respectively. Given this, the BLP problem is the equation (2.6):
(BLP I)
S y x
dy cx y x f
y by
ax y x F
y x
) , ( s.t.
, max
solves where
, max
(2.6)
The constraint set S {(x,y):AxByu,(x,y0)} is assumed to be a bounded, nonempty subset of Rn1n2.
Since S is assumed bounded and nonempty, for each x , the follower planner’s problem,
0
, s.t.
, max
y
x A u By
dy y x f
y
has an optimal solution. The set of all optimal solutions with respect to this x is called the feasible reaction set for the follower planner and is denoted Y
x . The leader planner’s feasible region, also referred to as the set of all rational reactions of f over S, is defined as)}
( ,
) , ( : ) , {(
)
(S x y x y S yY x
.
a point (x,y)(S) such that axbyaxby for all (x,y)(S) is called an optimal solution of the BLP.
The geometric properties of BLP problem have been well explored, and the pertinent ones are presented below:
(a) (S) is a connected subset of S.
(b) If there is an optimal solution to the BLP problem, then there is an extreme point of )
(S that is an optimal solution of the BLP problem, and hence there is an extreme point of S that is an optimal solution of the BLP problem.
The following theorem provides a geometric characterization of an optimal solution of a BLP problem.
Theorem 1. If an optimal solution of the leader objective function over S is in (S), then it is an optimal solution to the BLP problem.
Theorem 2. If there exists an optimal solution of the leader objective function over S not in (S), there exists a boundary feasible extreme point that optimizes the BLP problem.
Example 1:
0 , 0
12 4
12 2
3 s.t.
max
solves where
3 max
y x
y x
y x
y x y
y y
x
y x
The illustration of the method is presented in Fig. 2.4. It is easy to see that (S) is the set of points on line segments connecting (0, 0), (3, 0), and (4, 4), while the optimal solution of this BLP problem is the point (4, 4), which is a boundary feasible extreme point.
Figure 2.4 A Optimal Solution to the Linear BLP
Corresponding to the equation (2.6) and example 1, Bard (1998) [26] gave the following basic definition for a linear BLP solution:
(a) Constraint region of the BLP problem:
, : , , , , 0
x y x X y Y Ax By u x y
S .
(b) Feasible set for the follower for each fixed xX:
y Y By u Ax
x
S( ) :
(c) Projection of S onto the leader’s decision space:
x X y Y By u Ax
X
S( ) : ,
(d) Follower’s rational reaction set for xS( X):
( ,ˆ):ˆ ( )
( ): ( , ) ( ,ˆ), ˆ ( )
max arg where
) ˆ (
: ˆ) , ( max arg :
) (
x S y y x f y x f x S y x S y y x f
x S y y x f y
Y y x P
(e) Inducible region:
(x,y):(x,y) S,y P(x)
IR
The rational reaction set P(x) defines the response while the inducible region IR represents the set over which the leader may optimize his objective. Thus, in teams of the above notations, the linear BLP problem can be written as:
F(x,y):(x,y)IR
max
(0,0) (3,0) (0,3)
(4,4) (2,5)
A boundary feasible extreme point
S
To ensure that Fig. 2.5 has an optimal solution, Bard (1998) [26] gave the following assumption:
(a) S is nonempty and compact.
(b) For all decisions taken by the leader, the follower has some room to respond, i.e.,
) (x
P .
(c) P(x) is a point-to-point map.
Figure 2.5 Illustration of BLP Solution by using Definition
2.3.2 Discrete Variables
In many optimization problems, a subset of the variables is restricted to only take on discrete value. This can complicate the problem. To specify the model, let x1 be an n1-dimensional vector of continuous variables and x2 be an n2-dimensional vector of discrete variable, where x=(x1,x2) and n=n1+n2. Similarly, define y1 as an m1-dimensional vector of continuous variables and y2 as an m2-dimensional vector of discrete variable, where y=(y1,y2) and m=m +m . This leads to
Y
X
S
S(x)
S(X)
IR
f
F
P(x)
ˆ )
ˆ ,
( x y
(BLP II)
where all vectors and matrices are of the conformal dimension, and the linear terms in x have been omitted from the follower’s objective in function (2.7) [26] (Bard, 1998).
Note that it may be desirable to explicit include additional restrictions, such as upper and lower bounds, on the variables. In this case, let xX {x:l1j xj u1j, j1,2,...,n} and
Bard has investigated the properties of the zero-one linear BLP problem when some or all variables are restricted to binary values. Based on the specific instances of (2.7), it will be convenient to consider the problem in the form of (2.6) without reference to which variables are continuous and which are discrete; i.e.,
integer
For each xX, it will be assumed that the optimal solution of the lower level problem is unique. Along with the linear bilevel programming problem (L-BLPP) where X=Rn and Y=Rm, there are three models as shown below:
(a) Discrete linear bilevel programming (DL-BLP) problem, where4 X=Bn and Y=Bm; (b) Discrete-continuous linear bilevel (DCL-BLP) problem, where X=Bn and Y=Rm; and (c) Continuous-discrete linear bilevel (CDL-BLP) problem, where X=Rn and Y=Bm.
Figure 2.6 depicts the inducible regions associated with the four problems.
Figure 2.6 Inducible Regions for Versions of the Linear BLPP
Source: Bard Jonathan F., ―Practical Bilevel Optimization: Algorithms and Applications‖, Kluwer Academic Publishers, Netherlands, pp.235, 1998.
x y
L-BLPP
x y
DL-BLPP
x y
DCL-BLPP
x y
CDL-BLPP
Corresponding to Fig. 2.6, Bard (1998) [26] presented some properties and theorems as shown below:
Property 1: If SU Rnm, then IR is nonempty if S . If SU Rnm, then IR is nonempty if there exists a xX such that (x,y)SU.
Property 2: The inducible regions of DCL-BLPP and DL-BLPP are included in the inducible regions of L-BLPP and CDL-BLPP, respectively.
Property 3: For the L-BLPP, let S be a bounded set, i.e., a polytope. If SU Rnm, then L-BLPP, DL-BLPP, and DCL-BLPP have an optimal solution if S. If
m n
U R
S , then L-BLPP, DL-BLPP, and DCL-BLPP have an optimal solution if exists a xX such that (x,y)SU.
Theorem: Let SU Rnm, S and suppose there exists an optimal solution (x*,y*) to CDL-BLLP. Then (x*,y*) is a boundary (bd) point of S.
Algorithms designed to solve integer programs generally rely on some separation, relaxation, and understanding to construct ever-tighter bounds on the solution.