A note on the definition of a linear bilevel
programming solution
Charles Audet
1,2
, Jean Haddad 2 , Gilles Savard 1,2
Abstract
An alternative definition of the linear bilevel programming problem BLP has recently been proposed by Lu, Shi, and Zhang. This note shows that the proposed
definition is a restriction of BLP . Indeed, the new definition is equivalent to transferring the first-level constraints involving second-level variables into the second
level, resulting in a special case of BLP in which there are no first-level constraint
involving second-level variables. Thus, contrary to what is stated by the authors who
suggested the new definition, this does not allow to solve a wider class of problems,
but rather relaxes the feasible region, allowing for infeasible points to be considered
as feasible.
Key words: Optimization, Linear bilevel programming
1
Introduction
The aim of this note is to show that the deficiency pointed out in [8,9,10] is
not really a deficiency and that the new definition proposed is equivalent to
moving the first-level constraints involving the second-level variables into the
second level, which changes the nature of the problem.
A bilevel program is a program in which a subset of the variables is required to
be an optimal solution of a second mathematical program [1,2,3,5,6,7,12]. The
linear bilevel program is a special case in which all the constraints and the
Email addresses: [email protected] (Charles Audet ),
[email protected] (Jean Haddad), [email protected] (Gilles
Savard).
URL: http://www.gerad.ca/Charles.Audet (Charles Audet ).
1 GERAD
2 Département de Mathématique et de Génie Industriel, École Polytechnique de
Montréal, C.P. 6079, succ. Centre-ville, Montréal (Québec), H3C 3A7 Canada
Preprint submitted to Elsevier Science
3rd October 2005
objective functions of both programs are linear. An overview of linear bilevel
programming can be found in the textbook [4]. The linear bilevel program can
be formulated as:
min
ct x + dt y
x,y
s.t. x ∈ X
(BLP )
(x, y) ∈ P 1
y ∈ arg min
at w
w
w ∈ SBLP (x),
s.t.
where x, c ∈ Rnx , y, d, w, a ∈ Rny . X ⊆ Rnx and P 1 ⊆ Rnx +ny are polyhedral
sets, and SBLP (x) = {w : Bw ≥ b − Ax} ⊆ Rny with A ∈ Rm×nx , B ∈ Rm×ny ,
b ∈ Rm .
Let P 2 = {(x, y) : Ax + By ≥ b} ⊆ Rnx +ny be the polyhedra defined by the
second-level constraints. Thus, SBLP (x) corresponds to the projection of P 2
on the y-space for a given x.
Let us define the following sets:
n
o
S = (x, y) : x ∈ X, (x, y) ∈ P 1 ∩ P 2 ,
n
o
MBLP (x) = arg min at w : w ∈ SBLP (x) ,
w
(1)
IRBLP = {(x, y) ∈ S : y ∈ MBLP (x)} .
S is the polyhedra defined by the intersection of both the first-level and the
second-level constraints. MBLP (x) corresponds to the set of optimal solutions
for the second-level program for a given x. Finally, IRBLP , called the induced
(or inducible) region, represents the feasible region of BLP . With these definitions, BLP is equivalent to the problem
min
{ct x + dt y : (x, y) ∈ IRBLP }
x,y
.
In the presence of first-level constraints involving the second-level y variables,
IRBLP is not necessarily a connected set, it may even be discrete [12,1]. However, in the case where there are no first-level constraint involving y, then
IRBLP is always connected [12].
Many algorithms have been developed to solve BLP . The special case where
there are no constraints involving the y variables in the first-level has also been
studied, and it is important to note that transferring a first-level constraints
2
into the second-level is not equivalent to the original problem, as illustrated
by the following example (see figure 1):
(BLP2 )
(BLP1 )
max
x
max
x
s.t.
0≤x≤1
s.t.
0≤x≤1
x,y
x,y
y ∈ arg max
w
w
y=0
y ∈ arg max w
s.t.
w
s.t.
w ≤ x,
w≤x
w = 0.
Both problems contain the same constraints, except that the first-level cony
BLP1
y
(0,1)
BLP2
(0,1)
y=x
x=1
y=x
Second−level objective
First−level objective
11111111111111111111
00000000000000000000
00000000000000000000
11111111111111111111
00000000000000000000
11111111111111111111
(1,0)
0000
1111
0000
1111
0000 S
1111
x=1
Second−level objective
First−level objective
x
x
(1,0)
IR
IR = S
Figure 1. The effects of moving the constraint y = 0 from the first level in BLP1 to
the second level in BLP2 .
straint y = 0 in BLP1 is moved to the second level in BLP2 . It follows that
S = {(x, 0) : 0 ≤ x ≤ 1} for both problems, and for any x ∈ R we have
MBLP1 (x) = {x} ,
IRBLP1 = {(0, 0)} ,
MBLP2 (x) =


 {0} if x ≥ 0

∅
else,
IRBLP2 = {(x, 0) : 0 ≤ x ≤ 1} .
For BLP1 , the only feasible point and optimal solution is (0, 0) with objective function value 0. For BLP2 , the optimal solution is (1, 0) with objective
function value 1.
3
2
A restrictive definition for BLP
Recently, [8,9,10] have proposed an alternative definition for the solution of
BLP . They motivate their new definition by giving an instance of BLP for
which no solution can be found even if S is not empty. They then show that
this instance must have a Pareto optimal solution [10]. They claim that the
fact that a Pareto optimal solution exists but could not be found using the
actual definitions is a deficiency of the theory. In order to obtain this Pareto
optimal solution, they propose a new definition for the solution of BLP . They
state that this new definition “can solve a wider class of problems than current
capabilities permit”.
Let us first state the definitions of [8,9,10]. Let S be as above and define:
new
SBLP
(x) = {y : (x, y) ∈ S} ,
n
o
new
new
MBLP
(x) = arg min at w : w ∈ SBLP
(x) ,
w
(2)
new
new
IRBLP
= {(x, y) ∈ S : y ∈ MBLP
(x)} .
The difference between definitions (1) and (2) is that (2) implies that the second level is now responsible for respecting the first-level constraints involving
the y variables.
Let us first note that the solution of the original BLP , as it is strictly defined,
does not have to be Pareto optimal [11], and this is due to the intrinsic noncooperative nature of the model. So the fact that no Pareto optimal solution
could be found for the example presented in [10] is not really a deficiency: no
solution was found because no solution exists, and this may happen even if S
is not empty.
Let us consider the instance BLP1 . According to the new definition, the optimal solution is (1, 0) and this does not solve BLP1 as it is stated. Indeed, if
x = 1, then the constraint
y ∈ arg max w
w
s.t.
w≤x
implies y = 1, but this contradicts the first-level constraint y = 0, so that this
solution is not feasible.
Using definition (2) implies that the second-level also takes responsibility for
satisfying the first-level constraints involving the y variables. This is equivalent
to simply moving the constraint (x, y) ∈ P 1 from the first to the second level,
and this is a relaxation of problem BLP but still NP-hard [12].
4
3
An equivalence
This section formally shows that the new BLP definition is equivalent to
moving the first-level constraints to the second level. Define:
min
ct x + dt y
x,y
s.t. x ∈ X
(LSZ)
(x, y) ∈ P 1
y ∈ arg min
at w
w
s.t.
new
w ∈ SLSZ
(x),
to be the bilevel formulation of BLP proposed in [8,9,10], and
min
ct x + dt y
x,y
(BLP 0 )
s.t. x ∈ X
y ∈ arg min at w
w
s.t.
w ∈ SBLP 0 (x),
to be the bilevel problem in which the first-level constraints involving y variables (P 1 ) are moved to the second level, where
new
SLSZ
(x) = {y : (x, y) ∈ S} ,
SBLP 0 (x) = {y : (x, y) ∈ P 1 ∩ P 2 } .
Define IRLSZ to be the induced region of LSZ, and IRBLP 0 to be the induced
region of BLP 0 .
Theorem 1 IRLSZ = IRBLP 0 .
Proof: By definition,
IRBLP 0 = {(x, y) ∈ S : y ∈ MBLP 0 (x)},
new
IRLSZ = {(x, y) ∈ S : y ∈ MLSZ
(x)},
where S is the same in both induced regions since it is defined to be the
intersection of the first and second-level constraints. We need to show that
5
the second-level optimal sets are identical for any x ∈ Rnx .
new
new
(x)}
MLSZ
(x) = arg min{at w : w ∈ SLSZ
w
= arg min{at w : (x, w) ∈ S}
w
= arg min{at w : w ∈ SBLP 0 (x)}
w
= MBLP 0 (x).
Define IR to be the induced region of these two problems, that is, IR =
IRLSZ = IRBLP 0 . The next corollaries follow trivially:
Corollary 2 (x∗ , y ∗ ) is optimal for LSZ ⇔ (x∗ , y ∗ ) is optimal for BLP 0 .
Corollary 3 The induced region IR is connected.
Corollary 4 If S is nonempty and bounded, IR is nonempty and bounded
and an optimal solution (x∗ , y ∗ ) is attained at an extreme point of IR.
Thus, solving LSZ is equivalent to solving BLP 0 , which is not equivalent to
the original BLP since BLP 0 is a relaxation of BLP .
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