BLP relaxations - Integer Bilevel Linear Programming Problems: New Results and Applications

Figure 2.4. A simple instance for a bilevel hazmat transportation problem

2.2 BLP relaxations

2.2.1 Relaxation via removal of constraints

The different impact of upper level and lower level constraints on the bilevel formulation has to be

taken into account when we want to relax a BLP by dropping a subset of constraints. Unlike classical

mathematical programming models, the new formulation obtained removing one or more constraints

does not necessarily provide a valid relaxation. Indeed, this depends on the choice of the constraints

we remove. Let us consider the following problem:

min

x,y

−x + 3y

s.t.

y ≥ 1

y ∈ argmin

y

s.t.

x + y ≤ 4

x + y ≥ 2

x − y ≥ −1

x − y ≤ 1

The optimal solution of the BLP is (2, 1), which is vertex C of S, and the optimal value is 1, see Figure

2.5(a). If the upper level constraints y ≥ 1 is dropped, as depicted in Figure 2.5(b), the inducible

region is wider, and the new optimal solution is vertex E, (1.5, 0.5), and the objective function’s value

is 0. Finally, if the follower’s constraint x − y ≤ 1 is removed, the follower’s feasible set is larger, but

the new rational solutions of the follower violate the upper level constraint. The inducible region is

reduced to segment A–B and the optimal solution is vertex B, (1, 1), which gains an objective func-

tion equals 2. Hence, in the second reformulation represented in Figure 2.5(c), dropping a follower’s

constraint we do not obtain a valid relaxation and the optimal solution is the worst out of the three cases.

(a) Inducible region of the original problem (b) Removal of an upper level constraint

2.2 BLP relaxations 27

This result, which is apparently unclear, can be easily understood if we think to the inherent meaning

of bilevel programming problems. The general framework of bilevel programming is the presence of

two non cooperating decision makers. Dropping a lower level constraint means that the set of possible

choices for the follower is increased and thus he acquires more contractual power towards the leader.

This implies that the solutions space of the leader may reduce and the value of the optimal solution

may get worse. On the contrary, if an upper level constraint is removed, the solutions space of the

follower does not change, but a larger set of rational solutions may be considered acceptable by the

leader. Thus the value of the optimal solution can not be worse and this represent a valid relaxation.

The following proposition sums up the above mentioned results.

Proposition 2. A BLP can be correctly relaxed dropping a subset of constraints, if and only if these

are constraints under control of the leader.

2.2.2 Single level relaxation

One of the most used and immediate relaxation of a BLP is the so called single level relaxation. It

consists of dropping the follower’s objective function, turning the bilevel structure into a single level

one and assuming that only one decision maker is involved; from a mathematical point of view the

leader’s feasible set is fully defined by a set of constraints and it is not necessary to solve an inner

problem to check feasibility of a solution. The feasible set S does not change and the problem consists

of solving the leader’s objective function on S.

It is immediate to understand that the single level version of a BLP is a valid relaxation, as a

solution that is bilevel–feasible for the BLP is also feasible for the single level problem, but the contrary

may not be true.

There is a special case in which the optimal solution of a BLP and of the optimal solution of its

single level version coincides. For the sake of simplicity, let us assume that there are no upper level

constraints. If the following holds

∇

F (x, y) = ∇f (y)

it means that the leader’s and the follower’s objective function have the same verse, thus minimizing

(or maximizing) F (x, y), f (y) is minimized (or maximized) as well. In this case, the optimal solution

found solving the single level formulation is also optimal for the BLP.

This result may not hold if we introduce an upper level constraint which depends on the upper

level variables x. We now prove the following two theorems, which are necessary and sufficient

conditions under which the optimal solution of the single level relaxation of a BLP is also optimal for

the original problem. According to these conditions, it is easy to know whether the optimal solution of

the relaxation is bilevel–feasible, without solving the inner problem. We assume that S is compact and

non empty.

Theorem 10. If the optimal solution (¯x, ¯y) of the single level relaxation of a BLP is optimal for the

original BLP, then∇

F (x, y) = ∇f (y) and among the active constraints at (¯x, ¯y) there is at least

one constraint under control of the follower.

Proof. Let us assume that (¯x, ¯y) is a vertex of S in which the only active constraints are under control

of the leader. Let us define S

= S \ {Cx + Dy ≤ e}. From the previous sections we know that

S ⊆ S

and that solving the single level relaxation of BLP on S

two cases may occur: a) we found a

solution (x

, y

) on the boundary of S

, b) the single level relaxation does not admit a finite solution,

i.e. S

is unbounded. In case a), if (x

, y

) ≡ (¯x, ¯y), it means that there is at least an active constraint at

(¯x, ¯y) which is not under control of the leader and this contradicts our initial assumption. It follows that

(¯x, ¯y) is an internal point of S

, thus there exists a rational solution (¯x, y

∗

) such that f (y

∗

) < f (¯y). In

case b), once again, (¯x, ¯y) is an internal point of S

and ¯y is not a rational solution at ¯x. In both cases

(¯x, ¯y) is not bilevel–feasible, hence the proof.

2 Theorem 11. Given the optimal solution (¯x, ¯y) of the single level relaxation of a BLP, if ∇

F (x, y) =

∇f (y) and among the active constraints at (¯x, ¯y) there are no upper level constraints, (¯x, ¯y) is also

the optimal solution of BLP.

Proof. If there are no upper level constraints active at (¯x, ¯y), it means that if we solve the single level

relaxation on S

, defined as above, the optimal solution found is always (¯x, ¯y). Thus (¯x, ¯y) is a rational

solution and satisfies the upper level constraints, hence it is bilevel–feasible and optimal for the BLP,

which completes the proof.

2 In Figures 2.6(a)-(d) there is a graphic explanation of Theorems 10 and 11. In each figure we reported

both the feasible set S and some additional upper level constraints.

In Figure 2.6(a) an upper level constraint intersects the inducible region and solution A is a vertex of

S in which a lower level constraint is active; A is bilevel–feasible and is the optimal solution of the

BLP. In Figure 2.6(b), although a lower level constraint is active at A, unlike the previous case the

solution is not bilevel–feasible and the optimal solution of the BLP is vertex B: this shows that the

condition stated in Theorem 10 is only necessary and not sufficient. In Figure 2.6(c) an upper level

constraint intersects the inducible region but this does not change the optimal solution of the single

level relaxation problem and Theorem 11 holds. Finally, in Figure 2.6(d) all the constraints active at

vertex A are upper level constraints and by Theorem 11 the solution is not optimal for the BLP.

In document Integer Bilevel Linear Programming Problems: New Results and Applications (Page 43-46)