In this section, we derive a theoretical scalarization problem which gives us a similar
correctness property as for the case with two objectives in Theorem 4.8.
In the coverage-based Box-Algorithm in Section 5.3, for the used scalarization problems, we have to use lower bounds in the constraints to ensure that the new point lies in the current box. However, we have seen in Remark 5.13 that the new point does not have to be nondominated since it can be dominated from a point in another box.
Hence, for a given box B(`, u) with Y ∩(u−R3=) compact (cf. Lemma 5.3) and εi = `i+ui
2 for i = 1, 2, we desire a more suitable scalarization method which can be used either
• to find a point x∗∈ XE with f (x∗) ∈ B(`, u)(ε1,ε2,u3) or • to prove that there does not exist such a point.
In other words, it is desirable to solve the following problem efficiently:
( ˆPε11,ε2) lex min (f3(x), f2(x), f1(x)) s. t. x ∈ XE
f (x) ∈ B(`, u)(ε1,ε2,u3)
This formulation ( ˆPε11,ε2) has the disadvantage that the efficient set is assumed to be known in advance, which contradicts the fact that a representative system of YN should be computed.
However, ignoring the lexicographic objective function, we can express the condition
x ∈ XE implicitly with the help of a bilevel optimization problem2, in which the lower-level problem checks whether the current point is nondominated. This check is based on Benson’s method (cf. Ehrgott, 2005). A similar idea for checking efficiency of a proper face for linear problems was used in Sayın (1996). Here, this idea is incorporated for general problems in the lower-level of the following scalarization problem.
( ˆPε1,bilevel1,ε2 ) min x,zx z x (5.4a) s. t. x ∈ X (5.4b) f (x) ∈ B(`, u)(ε1,ε2,u3) (5.4c) zx := max ˜ x 3 X i=1 fi(x) − fi(˜x) (5.4d) s. t. ˜x ∈ X (5.4e) f (˜x) 5 f (x) (5.4f)
( ˆPε1,bilevel1,ε2 ) defines a so-called min-max problem, a special case of a bilevel optimization problem for which the lower-level and upper-level objective functions coincide.
In this context, we say 3OP is discrete, if it has three integer-valued objective functions, i.e., f : X −→ Z3. Hence, if we additionally assume that 3OP is discrete, we can also
add implicitly the minimization of the third objective function: ( ˆPIP,ε1,bilevel 1,ε2) minx,zx z x+ δ · f 3(x) (5.5a) s. t. x ∈ X (5.5b) f (x) ∈ B(`, u)(ε1,ε2,u3) (5.5c) zx := max ˜ x 3 X i=1 fi(x) − fi(˜x) (5.5d) s. t. ˜x ∈ X (5.5e) f (˜x) 5 f (x) (5.5f) where 0 < δ < 2·max{|`1 3|,|u3|} and `3< u3.
We first note some obvious but nice properties of these scalarization problems. Observation 5.39:
• If X is induced by linear constraints and f is a linear function, then all constraints and the objective function remain linear but now with bilevel structure.
• All terms of the sum in (5.4d) and (5.5d) are non-negative.
• The absolute value of the second summand δ · f3(x) in the objective function (5.5a) fulfills:
|δ · f3(x)| = δ · |f3(x)| ≤ δ · max{|`3|, |u3|} < 1 2 • For a feasible solution ˆx of ( ˆPIP,ε1,bilevel
1,ε2), it holds z
ˆ x ∈ N
0 since for all x ∈ X it
holds f (x) ∈ Z3 and zx≥ 0. C
For these two scalarization methods, we are now able to give a similar result as in Theorem 4.8.
Theorem 5.40 (Correctness Property): ( ˆPε1,bilevel1,ε2 ) is infeasible or it exists an op-
timal solution x∗∈ X with zx∗ > 0 if and only if B(`, u)(ε1,ε2,u3)∩ Y
N = ∅.
If 3OP is discrete, the statement also holds for ( ˆPIP,ε1,bilevel1,ε2).
Proof:
We only consider the case for ( ˆPIP,ε1,bilevel
1,ε2) since the proof for ( ˆP
1,bilevel
ε1,ε2 ) works analogously.
“⇒” :
Suppose there exists some x∗ ∈ XE with f (x∗) ∈ B(`, u)(ε1,ε2,u3) ∩ Y
(x, ˜x, zx) := (x∗, x∗, 0) is a feasible solution for ( ˆPIP,ε1,bilevel
1,ε2) with z
x∗
= 0. Further- more, no other feasible solution ˆx ∈ X with zxˆ > 0 (i.e., zxˆ ≥ 1) can obtain a better objective value since the second term of the objective function always fulfills
δ · f3(ˆx) ∈ −12,12. “⇐” : Let B(`, u)(ε1,ε2,u3)∩ Y N = ∅.
Case 1: B(`, u)(ε1,ε2,u3)∩ Y = ∅. Then, ( ˆP1,bilevel
IP,ε1,ε2) is infeasible.
Case 2: Let x∗ ∈ X with f (x∗) ∈ B(`, u)(ε1,ε2,u3)∩ Y \ Y
N be arbitrary. Then, there exists some x0 ∈ X with f (x0) ≤ f (x∗). Thus, the associated zx∗ fulfills zx∗ ≥ 1 > 0. Corollary 5.41: B(`, u)(ε1,ε2,u3)∩ Y
N 6= ∅ if and only if ( ˆPε1,bilevel1,ε2 ) is feasible and all
optimal solutions x∗ ∈ X have zx∗ = 0.
If 3OP is discrete, the statement also holds for ( ˆPIP,ε1,bilevel
1,ε2).
Theorem 5.42: Let 3OP be discrete. If there exists an optimal solution x∗ ∈ X for ( ˆPIP,ε1,bilevel
1,ε2) with z
x∗ = 0, then this solution fulfills:
(i) x∗∈ XE
(ii) x∗= arg minnf3(x) : x ∈ XE and f (x) ∈ B(`, u)(ε1,ε2,u3)
o
In particular, if 3OP is not discrete, (i) also holds for an optimal solution x∗ for
( ˆPε1,bilevel1,ε2 ).
Proof:
Let x∗ be an optimal solution for ( ˆPIP,ε1,bilevel1,ε2) (or ( ˆPε1,bilevel1,ε2 ), respectively) with zx∗ = 0. (i):
Suppose x∗ ∈ X/ E. Then, there exists some x0 ∈ X with f (x0) ≤ f (x∗). Thus, the associated zx∗ fulfills zx∗> 0 which contradicts the assumption.
(ii):
Suppose there exists a x0 ∈ XE with f (x0) ∈ B(`, u)(ε1,ε2,u3) and f
3(x0) < f3(x∗). Thus, the associated zx0 fulfills zx0 = 0 and it holds
δ · f3(x0) < δ · f3(x∗)
which contradicts optimality of x∗ for ( ˆPIP,ε1,bilevel
1,ε2).
We conclude this section with a remark about the practicability of the scalarization problems ( ˆPε1,bilevel1,ε2 ) and ( ˆPIP,ε1,bilevel
1,ε2).
Remark 5.43:
Due to its bilevel structure, problem ( ˆPε1,bilevel1,ε2 ) is hard to solve. Since such problems should be solved repeatedly in each iteration in the Box-Algorithm for three objectives,
we decided to investigate the simpler subproblem (Pε1
1,ε2) further, which can be solved
using standard black-box solver (like CPLEX). However, in the literature, there exist some methods to tackle this min-max problem for special assumptions on the constraints and objectives (see, e.g., Falk, 1973; Shimizu and Aiyoshi, 1981; Belenky, 1997), but, through the bilevel structure and the doubled constraints due to the introduction of the variable ˜x, no method with such an efficiency we need. For instance, Falk (1973)
considers the case with linear constraints and continuous variables max
x miny {c >
x + d>x : Ax + By 5 b, x, y = 0} . (5.6)
He first showed that an optimal solution must be located at a vertex of the correspond- ing polyhedron and developed a branch-and-bound algorithm solving this problem. Nevertheless, the algorithm is very time-consuming since the branching is done directly on the variables forcing them to be not in the basis. Hence, in the branch-and-bound scheme, for each father node, m many child nodes are obtained, where m denotes the rank of the constraint matrix. Since in the special subproblem ( ˆPε1,bilevel1,ε2 ), the constraints are doubled, this method and also other methods are nice from a theoretical point of view but are expected to be not efficient.
Despite the nice theoretical results for problem ( ˆPIP,ε1,bilevel
1,ε2), we are not aware of a method
solving such integral bilevel optimization problems in reasonable time. Theoretically, one can ask if methods as, e.g., the approach of Falk (1973) can be used to develop a branch-and-bound algorithm for bilevel integer linear problems. However, the linear relaxation (5.6) of the following integer linear max-min problem
max x miny {c
>x + d>
x : Ax + By ≤ b, x, y ∈ N0} (5.7)
cannot serve neither as lower nor as upper bound for the integer case problem, which can be observed in the following example:
We consider two different max-min problems. The first problem is given by max
x miny {y : 2y − x ≥ 0, 2y + x ≤ 6, x, y ∈ N0} . (5.8) The optimal solution is attained at (x, y) = (2, 1) with objective value 1. The optimal solution of the corresponding linear problem, where the constraints x, y ∈ N0 are substituted with x, y ≥ 0, is attained at (x, y) = (3, 1.5) with objective value 1.5. The second problem is given by
max
x miny {y : 2y − x ≥ 0, 2y + x ≤ 7, x ≤ 3, x, y ∈ N0} . (5.9) Here, the optimal solution is attained at (x, y) = (3, 2) with objective value 2. The optimal solution of the corresponding linear problem, where the constraints x, y ∈ N0 are substituted with x, y ≥ 0, is again attained at (x, y) = (3, 1.5) with objective value