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Chapter 1

BILEVEL PROGRAMMING:A COMBINATORIAL PERSPECTIVE

Patrice MarcotteDIRO and CRT, Universite de Montreal

[email protected]

Gilles SavardMAGI and GERAD, Ecole Polytechnique de Montreal

[email protected]

Abstract Bilevel programming is a branch of optimization where a subset of vari-ables is constrained to lie in the optimal set of an auxiliary mathemat-ical program. This chapter presents an overview of two specific classesof bilevel programs, and in particular their relationship to well-knowncombinatorial problems.

1. IntroductionIn optimization and game theory, it is frequent to encounter situations

where conflicting agents are taking actions according to a predefined se-quence of play. For instance, in the Stackelberg version of duopolisticequilibrium [Sta52], a leaderfirm incorporates within its decision processthe reaction of the follower firm to its course of action. By extendingthis concept to a pair of arbitrary mathematical programs, one obtainsthe class ofbilevel programs, which allow the modeling of many decisionprocesses. The term bilevel programming appeared for the first timein a paper by Candler and Norton [CN77], who considered a multi-levelformulation in the context of agricultural economics. Since that time,

hundreds of papers have been dedicated to this topic. The reader inter-ested in the theory and applications of bilevel programming is referred

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to the recent books by Shimizu, Ishizuka and Bard [SIB97], Luo, Pangand Ralph [LPR96], Bard [Bar98] and Dempe [Dem02].

Generically, a bilevel program assumes the form

minx,y

f(x, y)

s.t. (x, y) X

y S(x),

where S(x) denotes the solution set of a mathematical program param-eterized in the vector x, i.e.,

S(x) = arg m iny

g(x, y)

s.t. (x, y) Y.

In this formulation, the leader is free, whenever the set S(x) does notshrink to a singleton, to select an element of S(x) that suits her best.This corresponds to the optimistic formulation. Alternatively, the pes-simistic formulation refers to the case where the leader protects herselfagainst the worst possible situation, and is formulated as

minx

maxy

f(x, y)

s.t. (x, y) X

y S(x).

The scope of this chapter is limited to the optimistic formulation. Thereader interested in the pessimistic formulation is referred to the paper[LM96] by Loridan and Morgan.

In many applications, the lower level corresponds to an equilibriumproblem that is best represented as a (parametric) variational inequal-ity or, equivalently, a generalized equation. We then obtain an MPEC(Mathematical Program with Equilibrium Constraints), that is expressedas1

1Throughout the paper, we assume that vectors on the left-hand side of an inner productare row vectors. Symmetrically, right-hand side vectors are understood to be column vectors.Thus primal (respectively dual) variables usually make up column (respectively row) vectors.Transpose are only used when absolutely necessary.

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Bilevel Programming: a Combinatorial Perspective 3

MPEC : minx,y

f(x, y)

s.t. (x, y) X

y Y(x)

G(x, y) NY(x)(y),

where Y(x) = {y : (x, y) Y} and NC(z) denotes the normal cone tothe set C at the point z. If the vector function G represents the gradientof a differentiable convex function g and the set Y is convex, then MPECreduces to a bilevel program. Conversely, an MPEC can be reformulatedas a standard bilevel program by noting that a vector y is solution ofthe lower level variational inequality if and only if it globally minimizes,with respect to the argument y, the strongly convex function gap(x, y)defined as (see Fukushima [Fuk92]):

gap(x, y) = maxyY(x)

G(x, y)(y y) 1

2y y2.

Being generically non-convex and non-differentiable, bilevel programsare intrinsically hard to solve. For one, the linear bilevel program whichcorresponds to the simple situation where all functions involved arelinear, is strongly N P-hard (see Section 1.2.2). Further, determiningwhether a solution is locally optimal is also strongly N P-hard [VSJ94].In view of these results, most research has followed two main avenues,either continuous or combinatorial. The continuous approach is mainlyconcerned with the characterization of necessary optimality conditionsand the development of algorithms that generate sequences convergingtoward a local solution. Along that line, let us mention works basedon the implicit function approach (Kocvara and Outrata [KO94]), onclassical nonlinear programming techniques such as SQP (SequentialQuadratic Programming) applied to a single-level reformulation of thebilevel problem (Scholtes and Stohr [SS99]) or smoothing approaches(Fukushima and Pang [FP99]) and Marcotte et al. [MSZ01]). Most workdone on MPECs follows that line.

The combinatorial approach takes a global optimization point of viewand looks for the development of algorithms with a guarantee of globaloptimality. Due to the intractability of the bilevel program, these al-gorithms are limited to specific subclasses possessing properties such as

linear, bilinear or quadratic objectives, which allow for the developmentof efficient algorithms. We consider two classes that are amenable toa global approach, namely bilevel programs involving linear or bilinearobjectives. The first class is important as it encompasses a large number

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4. For a given vector x 2x, the set of optimal solutions of the lowerproblem is

S(x) = {y : y arg max{d2y : y y(x)}.

A point (x, y) is said to be rational if x 2x S(x).

5. The optimal value function for x 2x is

v(x) = d2y, y S(x).

6. The admissible set (also called induced region) is

= {(x, y) : x 0, A1x + B1y b1, y S(x)} .

A point (x, y) is admissible if it is feasible and lies in S(x).

Based on the above notations, we characterize optimal solutions forthe LLBP.

Definition 1.2 A point (x, y) is optimal for LLBP if it is admissibleand, for all admissible (x, y), there holds c1x

+ d1y c1x + d1y.

Note that, whenever the upper level constraints involve no lower levelvariables, then rational p oints are also admissible. The converse mayfail to hold in the presence of joint upper level constraints.

To illustrate some geometric properties of bilevel programs (see Fig-ure 1.2), let us consider the following two-dimensional example (see also

Figure 1.1 (a):

maxx,y

x 4y

s.t. x 0

y arg maxy

y

s.t. 2x y 8

3x + 2y 6

5x + 6y 60

2x + y 162x 5y 0

y 0.

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The following theorem is a direct consequence of the polyhedral natureof the admissible set. It emphasizes the combinatorial nature of theLLBP.

Theorem 1.3 If LLBP has a solution, an optimal solution is attainedat an extreme point of .

The combinatorial nature of bilevel programming can also be ob-served by studying the single-level reformulation obtained by replacingthe lower level problem by its (necessary and sufficient) optimality con-ditions:

LLBP1 : maxx,y,

c1x + d1y

s.t. A1x + B1y b1A2x + B2y b2B2 d2(b2 A2x B2y) = 0(B2 d2)y = 0x 0, y 0, 0,

where Rnl . The combinatorial nature is entirely captured by the twoorthogonality constraints, which can actually be added to form a singleconstraint. Their disjunctive nature relates the LLBP to linear mixedinteger programming and allows for the development of algorithms basedon enumeration and/or cutting plane approaches.

2.1 Equivalence between LLBP and classicalproblems

In this section, we show that simple polynomial transformations al-low to formulate linear mixed 0-1 integer programs (MIP01) and bilin-ear disjoint programs (BDP) as linear bilevel programs, and vice versa.The interest in these reformulations goes beyond the complexity issue.Indeed, Audet [Aud97] and Audet et al. [AHJS97] have uncovered equiv-alences between algorithms designed to solve (MIP01) and LLBP. Theyhave shown that the HJ S algorithm of [HJS92] designed for solving theLLBP can be mapped onto a standard branch-and-bound method (seefor instance [BS65] for addressing an equivalent mixed 0 1 program,provided that mutually consistent branching rules are applied. We maytherefore claim that the mixed 0 1 algorithm is subsumed (the authorsuse the term embedded) by the bilevel algorithm. This result shows that

the structure of both problems is virtually indistinguishable, and thatany algorithmic improvement on one problem can readily be adaptedto the other [AHJS97]: solution techniques developed for solving mixed0 1 programs may be tailored to the LLBP, and vice-versa.

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2.1.1 LLBP and MIP01. The linear mixed 0 1 programmingproblem (MIP01) is expressed as

MIP01 : maxx,u cx + eus.t. Ax + Eu b

x 0, u binary valued,

where c Rnx , e Rnu, A Rmnx , E Rmnu , b Rm.

We first note that the binary condition is equivalent to:

0 u 10 = min{u, 1 u},

where 1 denotes the vector of all ones. Next, by introducing an up-per level variable y, and defining a second level problem such that theoptimal solution corresponds to this minimum, we obtain the equivalentbilevel programming reformulation:

LLBP2 : maxx,y,u

cx + eu

s.t. Ax + Eu b0 u 1x 0y = 0

y arg maxw

nui=1

wi

s.t. w uw 1 u.

where y, w Rnu. In this formulation, the integrality constraints are nomore required, as they are enforced by the upper level constraints y = 0,together with the lower level optimality conditions.

In general, upper level constraints make the problem more difficultto solve. Actually, some algorithms only address instances where suchconstraints are absent. However, as suggested by Vicente, Savard and

Judice [VSJ96], the constraint y = 0 can be enforced by incorporating anexact penalty within the leaders objective, i.e., there exists a thresholdvalue M such that, whenever M exceeds M, the solution of the fol-lowing bilevel program satisfies the condition y = 0, i.e., the integrality

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condition:

LLBP3 : maxx,y,u

cx + eu M1y

s.t. Ax + Eu b0 u 1x 0y arg max

w

nui=1 wi

s.t. w uw 1 u.

Conversely, LLBP may be polynomially reduced to a MIP01. First,one replaces the lower level problem by its optimality conditions, yieldinga single-level program with the complementarity constraints

(b2 A2x B2y) = 0(B2 d2)y = 0.

The second transformation consists in linearizing the complementarityconstraints by introducing two binary vectors u and v and a sufficientlylarge finite constant L > 0, the existence of which is discussed in [VSJ96)]:

b2 A2x B2y L(1 u), LuT,

y L(1 v) B2 d2 LvT.

This leads to the equivalent MIP01 reformulation ofLLBP:

MIPLLBP : maxx,y,,u,v

c1x + d1y

s.t. A1x + B1y b1x 0A2x + B2y b2 B2 d2y 0 0

A2x B2y + Lu L1 b2 LuT 0y + Lv L1 B2 LvT d2u binary valued v binary valued.

2.1.2LLBP

andBILP

. The disjoint bilinear programmingproblem BILP was introduced by Konno [Kon71] to generalize Mills ap-proach [Mil60] for computing Nash equilibra [Nas51] of bimatrix games.It can be expressed as follows:

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BILP : maxx,u

cx uQx + ud

s.t. Ax b1uB b2

x 0u 0,

where c Rnx , d Rnu, Q Rnunx, A Rnvnx , B Rnuny , b1 Rnv , b2 Rny , and the matrix Q assumes no specific structure.By exploiting the connection between LLBP and BILP, Audet et al.

[AHJS99] and Alarie et al. [AAJS01] have been able to construct im-proved branch-and-cut algorithms for the BILP. Their approach relieson the separability, with respect to the vectors x and u, of the feasibleset of BILP. Let us introduce the sets X = {x 0 : Ax b1} andU = {u 0 : uB b2}. If both sets are nonempty and the optimalsolution ofBILP is bounded, we can rewrite BILP as

BILP2 : maxxX cx + maxuU u(d Qx).

For fixed x X, one can replace the inner optimization problem by itsdual, to obtain

maxxX

cx + miny

b2y

s.t. Qx + By dy 0.

Under the boundedness assumption, the dual of the inner problem isfeasible and bounded for each x X. In a symmetric way, one canreverse the roles of x and u to obtain the equivalent formulation

maxuU ud + minv vb1s.t. uQ + vA c

v 0.

Thus, the solution ofBILP can be obtained by solving either one of thesymmetric bilevel programs

LLBP4

maxx,y

cx + b2y

s.t. Ax b1x 0

y arg miny b2ys.t. Qx + By d

y 0

LLBP5

maxu,v

ud + vb1

s.t. uB b2u 0

v arg minv vb1s.t. uQ + vA c

v 0,

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These two problems correspond to max-min programs, i.e., bilevelprogram involving opposite objective functions.

If BILP is unbounded, the above transformations are no longer valid

as the inner problem may prove infeasible for some values of x X (oru U). For instance, the existence of a ray (unbounded direction) inu-space implies that there exist x X and u with uB 0 such thatu(d Qx) > 0. Equivalently there exists a vector x such that the innerproblem in BILP2 is unbounded, which implies in turn that its dual isinfeasible with respect to x.

In order to be equivalent to BILP, LLBP4 should therefore select anx-value for which the lower level problem is infeasible. However, thisis inconsistent with the optimal solution of a bilevel program being ad-missible. Actually, Audet et al. [AHJS99] have shown that determiningwhether there exists an x in X such that

Y(x) = {y 0 : By d Qx}

is empty, is strongly N P-complete. Equivalently, determining if BILP isbounded is strongly N P-complete. This was achieved by constructingan auxiliary bilinear program BILP (always bounded) such that BILP isunbounded whenever the optimal value of BILP is positive. Based onthis technique, the bilevel reformulation can be used to solve separablebilinear programs, whether they are bounded or not.

2.2 Complexity of linear bilevel programming

While one may derive complexity results about bilevel programs viathe bilinear programming connection, it is instructive to perform directlyreductions from standard combinatorial problems. After Jeroslow [Jer85]initially proved that LLBP is NP-hard, Hansen, Jaumard and Savard [HJS92] showed NP-hardness, using a reduction from KERNEL (see Gareyand Johnson [GJ79]. Vicente, Savard and Judice [VSJ94] strengthenedthese results and proved that checking strict and local optimality arealso NP-hard. In this section, we present different proofs, based on areduction from 3-SAT.

Let x1, . . . , xn be n Boolean variables and

F =mi=1

(li1 li2 li3)

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be a 3-CNF formula involving m clauses with literals lij.3 To each clause

(li1 li2 li3) we associate a linear Boolean inequality of the form

vi1 + vi2 + vi3 1

where

vij =

xk if lij = xk,

1 xk if lij = xk.

According to this scheme, the inequality

x1 + (1 x4) + x6 1

corresponds to the clause (x1 x4 x6). Using matrix notation, theinequalities take the form

ASx 1 + c

where AS is a matrix with entries in {0, 1, 1}. By definition, S issatisfiable if and only if a feasible binary solution of this linear systemexists. We have seen that it is indeed easy to force variables to takebinary values trough a bilevel program. The reduction makes use of thistransformation.

Theorem 1.4 LLBP is strongly NP-hard.

Proof. Consider the following LLBP:

minx,z F(x, z) =

n

i=1 zi

subject to ASx 1 + c

0 xi 1, i = 0, . . . , n

z argmax {ni=1

zi :

zi xizi 1 xi

z 0}

We claim that S is satisfiable if and only if the optimal solution of

the LLBP is 0 (note that 0 is a lower bound on the optimal value).

3A literal consists in a variable or its negation.

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First assume that S is satisfiable and let x = (x1, . . . , xn) be a truthassignment for S. Then the first level constraints are verified and the solefeasible lower level solution corresponds to setting zi = 0 for all i. Since

this rational solution (x, z) achieves a value of 0, it is optimal. Assumenext that S is not satisfiable. Any feasible x-solution must be fractionaryand, since every rational solution satisfies zi = min{xi, 1 xi}, at leastone zi must assume a positive value, and the objective F(x, z) cannotbe driven to zero. This completes the proof. 2

Corollary 2.1 There is no fully polynomial approximation scheme forLLBP unless P = N P.

To prove the local optimality results, Vicente, Savard and Judice

adapted techniques developed by Pardalos and Schnitger [PS88] for non-convex quadratic programming, where the problem of checking (strictor not) local optimality was proved to be equivalent to solving a 3-SATproblem. The present proof differs slightly from the one developed in [VSJ94].

The main idea consists in constructing an equivalent but degeneratebilevel problem of 3-SAT. For that, we augment the Boolean constraintswith an additional variable x0, change the right hand-side to 3/2, andbound the x variables. For each instance S of 3-SAT, let us consider theconstraint set:

ASx + Ix0 3

2+ c

1

2 x0 xi

1

2+ x0, i = 1, . . . , n

xi 0, i = 0, . . . , n .

Obviously, the solution x = (0, 1/2, . . . , 1/2) satisfies the above linearinequalities, but this does not guarantee that S is satisfiable. Hence, wewill consider a bilevel program that will have, at this solution, the sameobjective value than we would obtain if S is satisfiable.

Theorem 2.1 Checking strict local optimality in linear bilevel program-ming is N P-Hard.

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Proof. Consider the following instance of a linear bilevel program:

minx,l,m,z

F(x,l,m,z) =n

i=1

zi

subject to ASx + Ix0 3

2+ c

1

2 x0 xi

1

2+ x0, i = 1, . . . , n

xi 0, i = 0, . . . , n

l,m,z argmax {ni=1

zi :

xi li =1

2 x0

xi

+ mi

=1

2+ x

0

zi li, zi mi, i = 1, . . . , n

z 0}

Let x = (0, 1/2, . . . , 1/2) and l = m = z = 0. We claim that S issatisfiable if and only if the point (x, l, m, z) is not a strict minimum.Since all variables zi are forced to be nonnegative then:

F(x,l,m,z) 0.

First, assume that S is satisfiable. Let x1, . . . , xn be a true assignmentfor S and set, for any x0 [0, 1/2]

x =

1/2 x0 if xi = 0,1/2 + x0 if xi = 1,

i.e., x satisfies the upper level constraints. Furthermore l = 0, m = 0and z = 0 is the optimal solution of the lower level problem for x fixed.Hence (x, l, m, z) belongs to the induced region associated with the linearbilevel program. Since F(x, l, m, z) = 0, we claim that (x, l, m, z) is aglobal minimum of the linear bilevel program.

Clearly, F(x,l,m,z) = 0 if and only if xi {12 x0,

12 + x0}, for all

i = 1, . . . , n. If this last condition holds, then li = 0 or mi = 0 andzi = 0 for all i = 1, . . . , n and F(x,l,m,z) = 0. Since x0 can be chosen

arbitrarily close to 0, x

cannot be a strict local minimum.Assume next that (x, l, m, z) is not a strict local minimum. Thereexists a rational point (x1, l1, m1, z1) such that F(x1, l1, m1, z1) = 0, andthis point satisfies l1 = m1 = z1 = 0 and x1 = 12 x0 or x

1 = 12 + x0 for

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all i and some x0. Then the assignment

xi = 0 if 1/2 x0,xi = 1 if 1/2 + x0

is a truth assignment for S. 2

Theorem 2.2 Checking local optimality in linear bilevel programmingis N P-hard.

The proof, which is based on complexity results developed in [PS88]and [VSJ94], will not be presented. Let us however mention that theunderlying strategy consists in slightly discriminating against the ratio-nal points assuming value 0, through the addition of penalty factor withrespect to x0, yielding the LLBP

minx,l,m,z,w

F(x,l,m,z,w) =ni=1

zi 1

2n

ni=1

wi

subject to ASx 3

2+ c

1

2 x0 xi

1

2+ x0, i = 1, . . . , n

xi 0, i = 0, . . . , n

l,m,z,w argmax {ni=1

zi ni=1

wi :

xi li = 12

x0

xi + mi =1

2+ x0

zi li, zi mi, i = 1, . . . , n

wi xi 1

2, wi

1

2 xi, i = 1, . . . , n

z, w 0.}

3. Optimal pricing via bilevel programming

Although much attention has been devoted to linear bilevel programs,

their mathematical structure does not fit many real life situations, whereit is much more likely that interaction between conflicting agents occursthrough the models objectives rather than joint constraints. In thissection, we consider such an instance that, despite its simple structure,

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forms the paradigm that lies behind large-scale applications in revenuemanagement and pricing, such as considered by Cote et al. [CMS03].

3.1 A simple pricing modelLet us consider a firm that wants to price independently (bundling

is not allowed) a set of products aimed at customers having specificrequirements and alternative purchasing sources. If the requirementsare related in a linear manner to the resources (products), one obtainsthe bilinear-bilinear bilevel program (BBBP):

BBBP : maxt,x,y

tx

s.t. (x, y) arg minx,y

(c + t)x + dy

Ax + By = b

x, y 0,

where t denotes the upper level decision vector, (c, d) the before taxprice vector , (x, y) the consumption vector, (A, B) the technology ma-trix and b the demand vector. In the above, a trade-off must be achievedbetween high t-values that price the leaders products away from the cus-tomer(s), and low prices that induce a low revenue.

In a certain way, the structure ofBBBP is dual to that ofLLBP, in thatthe constraint set is separable and interaction occurs only through theobjective functions. The relationship between LLBP and BBBP actuallygoes further. By replacing the lower level program by its primal-dualcharacterization, one obtains the equivalent bilinear and single-level pro-

gram

maxt,x,y

tx

s.t. Ax + By = b

x, y 0

A c + t

B d

(c + t A)x = 0

(d B)y = 0.

Without loss of generality, one can set t = A c. Indeed, if xi > 0,ti = (A)i ci follows from the next-to-last orthogonality conditionswhereas, if xi = 0, the leaders objective is not affected by the value ofti. Now, a little algebra yields:

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tx = Ax cx = (b By) cx = b (cx + dy)

and one is left with a program involving a single nonlinear (actuallybilinear and separable) constraint, that can be penalized to yield thebilinear program

PENAL : maxx,y,

b (cx + dy) M(d B)y

s.t. Ax + By = b

x, y 0

B d.

Under mild feasibility and compactness assumptions, it has been shownby Labbe et al. [LMS98] that there exists a finite value M of the penaltyparameter M such that, for every value of M larger than M, any op-timal solution of the penalized problem satisfies the orthogonality con-straint (d B)y = 0, i.e., the penalty is exact.4 Since the penalizedproblem is bilinear and separable, optimality must be achieved at someextreme point of the feasible polyhedron. Moreover, the program can,using the techniques of Section 1.2.1.2, be reformulated as a linear bilevelprogram of a special type.

The reverse transformation, from a generic LLBP to BBBP, is notstraightforward and could not b e achieved by the authors. However,

since BBBP is strongly N P-hard, such polynomial transformation mustexist.

3.2 Complexity

In this section, we consider a subclass ofBBBP initially considered byLabbe et al., where the feasible set {(x, y) : Ax+By = b,x,y 0} is thatof a multicommodity flow problem, without upper bound constraints onthe links of the network. For a given upper level vector t, a solutionto the lower level problem corresponds to assigning demand to shortestpaths linking origin and destination nodes. This yields:

4Be careful though: the stationary p oints of the penalized and original problems need not bein one-to-one relationship!

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TOLL : maxt,x,y

tkK

xk

s.t. (xk

, yk

) arg minxk,yk txk + dyk

Axk + Byk = bk

xk, yk 0

k K,

where (A, B) denotes the node-arc incidence matrix of the network, andbk denotes the demand vector associated with the origin-destination pair,or commodity k K.

Note that since a common toll vector t applies to all commodities,TOLL does not quite fit the format of BBBP. However, by settingx =

kKx

k for both objectives5 and incorporating the compatibility

constraint x =

kKxk (at either level), we obtain a bona fideBBBP.

Theorem 1.5 TOLL is strongly NP-hard, even when |K| = 1.

The proof relies on the reduction on the reformulation of 3-SAT astoll problem involving a single origin-destination pair, and is directlyadapted form the paper by Roch et al. [RSM04]. Let x1, . . . , xn be nBoolean variables and

F =mi=1

(li1 li2 li3) (1.1)

be a 3-CNF formula consisting of m clauses with literals (variables ortheir negations) lij . For each clause, we construct a cell, i.e., a sub-

network comprising one toll arc for each literal. Cells are connected by apair of parallel arcs, one of which is toll-free, and by arcs linking literalsthat cannot be simultaneously satisfied (see Figure 1.3).

The idea is the following: if the optimal path goes through toll arcTij, then the corresponding literal lij is true. The sub-networks areconnected by two parallel arcs, a toll-free arc of cost 2 and a toll arc ofcost 0, as shown in Figure 1.3.

If F is satisfiable, we want the optimal path to go through a singletoll arc per sub-network (i.e., one true literal per clause) and simultane-ously want to make sure that the corresponding assignment of variablesis consistent; i.e., paths that include a variable and its negation mustbe ruled out. For that purpose, we assign to every pair of literals cor-

responding to a variable and its negation an inter-clause toll-free arc

5This is allowed by the fact that the lower level constraints are separable by commodity.

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1 0 + T21 0 + T22 0 + T23

0

00

00

0

1 0 + T31 0 + T32 0 + T33

0

0 0 0

00

1 0 + T11 0 + T12 0 + T13

0 0

00 0

0

0 + T12

2 0 + T2

s

t

1

1

1 1

1

Figure 1.3. Network for the formula (x1 x2 x3) (x2 x3 x4) (x1 x3 x4).Inter-clause arcs are bold. Path through T12, T22, T32 is optimal (x2 = x3 =true).

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between the corresponding toll arcs (see Figure 1.3). As we will see, thisimplies that inconsistent paths, involving a variable and its negation,are suboptimal.

Since the length of a shortest toll-free path is m + 2(m 1) = 3m 2and that of a shortest path with zero tolls is 0, 3m 2 is an upper boundon the revenue. We claim that F is satisfiable if and only if the optimalrevenue is equal to that bound.

Assume that the optimal revenue is equal to 3m 2. Obviously, thelength of the optimal path when tolls are set to 0 must be 0, otherwisethe upper bound cannot b e reached. To achieve this, the optimal pathhas to go through one toll arc per sub-network (it cannot use inter-clausearcs) and tolls have to be set to 1 on selected literals, C + 1 on otherliterals and 2 on tolls Tk, k. We claim that the optimal path doesnot include a variable and its negation. Indeed, if that were the case,the inter-clause arc joining the corresponding toll arcs would impose a

constraint on the tolls between its endpoints. In particular, the toll Tkimmediately following the initial vertex of this inter-clause arc wouldhave to be set at most to 1, instead of 2. This yields a contradiction.Therefore, the optimal path must correspond to a consistent assignment,and F is satisfiable (note: if a variable and its negation do not appearon the optimal path, this variable can be set to any value).

Conversely if F is satisfiable, at least one literal per clause is true ina satisfying assignment. Consider the path going through the toll arcscorresponding to these literals. Since the assignment is consistent, thepath does not simultaneously include a variable and its negation, andno inter-clause arc limits the revenue. Thus, the upper bound of 3m 2is reached on this path.

Another instance, involving several commodities but restricting eachpath to use a single toll arc, also proved N P-hard. Indeed, consider theriver tarification problem, where users cross a river by either usingone of many toll bridges, or by flying directly to their destination on atoll-free arc. The proof of N P-completeness also makes use of 3-SAT,but there is a twist: each cell now corresponds to a variable rather than aclause, and is thus dual to the previous transformation (see Grigorievet al. [GvHvdK+04] for details). Apart of its elegance, the dual reductionhas the advantage of being related to the corresponding optimizationproblem, i.e., one can maximize the number of satisfied clauses by solvingthe related TOLL problem. This is not true of the primal reduction,where the truth assignment is only valid when the Boolean formula can

be satisfied. Indeed, the solution of the TOLL reduction may attain anear-optimal value of 3m 3 without any clause being satisfied, thusmaking the truth assignment of the variables irrelevant. For instance,

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consider an instance where a variable and its negation appear as literalsin the first and last clauses.6 Then, a revenue of 3m 3, one less thanthe optimal revenue, is achieved on the path that goes through the two

literals and the toll-free link between them, by setting the tolls on thetwo toll arcs of that path to 0 and 3m 3 respectively.We conclude this section by mentioning that TOLL is polynomially

solvable when the number of toll arcs is bounded by some constant. Ifthe set of toll arcs reduces to a singleton, a simple ordering strategy canbe applied (see [LMS98]). In the general case, path enumeration yieldsa polynomial algorithm that is unfortunately not applicable in practice(see Grigoriev et al. [GvHvdK+04]). Other polynomial cases have beeninvestigated by van Hoesel et al. [vHvdKM+03].

3.3 The traveling salesman problem

Although the relationship between the traveling salesman problem

(TSP in short) and TOLL is not obvious, the first complexity resultinvolved TSP or, to be more precise, the Hamiltonian path problem(HPP). The reduction considered in [LMS98] goes as follows: Givena directed graph with n nodes, among them two distinguished nodes:an origin s and a destination t the destination, we consider the graphobtained by creating a toll-free arc from s to t, with length dst = n 1.Next, we endow the remaining arcs, all toll arcs, with cost 1 and imposea lower bound of 2 on all of them. Then, it is not difficult to see thatthe maximal toll revenue, equal to 2n 2, is obtained by setting ta = 2on the arcs of any Hamiltonian path, and ta = n + 1 elsewhere.

The weakness of that reduction is that it rests on two assumptionsthat are not required in the reductions presented in the previous sec-tions, that is, negativity of arc lengths and lower bounds on toll values.Notwithstanding, the relationship between TOLL and TSP has provedfruitful. To see this, let us follow Marcotte et al. [MSS03] and considera TSP involving a graph G and a length vector c. First, we transformthe TSP into an HPP by duplicating the origin node s and replacingall arcs (i, s) by arcs from i to t. It is clear that the solutions to TSPand HPP are in one-to-one correspondence. Second, we incorporate atoll-free arc (s, t) with cost n, we set the fixed cost of the remaining arcsto 1 + ca/L and the lower bounds on tolls to 2 ca/L, where L issome suitably large constant, L = n maxa{ca} for instance. Then, any

6Remark: The clauses involving the two opposite literals can always be made the first andthe last, through a straightforward permutation. This shows that the model is sensitive tothe rearrangement of clauses.

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solution to the toll problem on the modified network yields a shortestHamiltonian path. This toll problem takes the form

maxt,x,ya

taxa

(x, y) arg minx,y

a

(1 + ca/L + ta)xa + nyst

s.t. flow conservationx 0.

Replacing the lower level linear program by its optimality conditions,one obtains a linear program including additional complementarity con-straints. The latter, upon the introduction of binary variables, can belinearized to yield a MIP formulation of the TSP that, after some trans-formations, yields:

minx,u

a

cijxij

s.t.j

xij = 1 i

i

xij = 1 j

ui uj (n 2) + (1 n)xij + (3 n)xji (i, j)

uj (n 2) + (3 n)x1j + xj(n+1) j = 1

uj (n 3)xj(n+1) x1j + 2 j = 1

x binary valued,

where u corresponds to the dual vector associated with the lower levelprogram. It is in a certain way surprising, and certainly of theoreticalinterets that, through standard manipulations, one achieves the mixedinteger program Note that this program is nothing but the lifted for-mulation of the Miller-Tucker-Zemlin constraints derived by Desrochersand Laporte [DL91], where the three constraints involving the vector uare facet-defining.

For complete directed graphs, the analysis supports a multicommodityextension, where each commodity is assigned to a subtour between twoprespecified vertices. More precisely, let [v1, v2, . . . , v|K|] be a sequenceof vertices. Then, the flow for commodity k K must follow a path

from vertex vk to vk+17, and the sequence of such paths must form

7By convention, vK+1 v1.

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a Hamiltonian circuit. If the number of commodity is 3 or less, theordering of the vertices is irrelevant. If |K| is more than 3, it is yetpossible to find a set of vertices that are extreme points of the convex

hull of vertices, together with the order in which they must be visited insome optimal tour (see Flood [Flo56]).When applied to graphs from the TSPLIB library [TSP], the linear

relaxation of the three-commodity reformulation provides lower boundsof quality comparable to those obtained by the relaxation proposed byDantzig, Fulkerson and Johnson [DFJ54]. This is all the more surprisingin the view that the latter formulation is exponential, while the formeris in O(n2).

3.4 Final considerations

This chapter has provided a very brief overview of two importantclasses of bilevel programs, from the perspective of combinatorial opti-

mization. Those classes are not the only ones to possess a combinatorialnature. Indeed, let us consider a bilevel program (or an MPEC) wherethe induced region is the union of polyhedral faces.8 A sufficient con-dition that an optimal solution be attained at an extreme point of theinduced region is then that the upper level objective be concave in bothupper and lower level variables. An interesting situation also occurswhen the upper level objective is quadratic and convex. In this case,the solution of the problem restricted to a polyhedral face occurs at anextreme point of the primal-dual polyhedron, and it follows that theproblem is also combinatorial.

Actually, bilevel programs almost always integrate a combinatorialelement. For instance, let us consider the general bilevel program:

minx,y

f(x, y)

s.t. y arg miny

g(x, y)

G(x, y) 0.

Under suitable constraints (differentiability, convexity and regularity ofthe lower level problem), one can replace the lower level problem by itsKuhn-Tucker conditions and obtain the equivalent program

8This situation is realized when the lower level is a linear, a convex quadratic, or a linearcomplementarity problem, and joint constraints, whenever they exist, are linear.

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BLKKT : minx,y

f(x, y)

s.t. G(x, y) 0

yg(x, y) + yG(x, y) = 0

G(x, y) = 0.

If the set of active constraints were known a priori, BLKKT would reduceto a standard nonlinear program. Provided that f, g and each of theGis be convex, the last constraint could yet make it non-convex, albeitweakly, in the sense that replacing all functions by their quadraticapproximations would make the bilevel problem convex. The main com-putational pitfall is actually the identification of the active set. Thistwo-sided nature of bilevel programming and MPEC is well capturedin the formulation proposed by Scholtes [Sch04], which distinguishes

between the continuous and combinatorial natures of MPECs. By re-arranging variables and constraints, one can reformulate BLKKT as thegeneric program

minx

f(x)

s.t. G(x) Z.

IfZ is the negative orthant, this is nothing more than a standard nonlin-ear program. However, special choices of Z, may force pairs of variablesto be complementary. It is then ill-advised to linearize Z, and the rightapproach is to develop a calculus that does not sidestep the combinato-rial nature of the set Z. Along that line of reasoning, Scholtes proposes

an SQP (Sequential Quadratic Programming) algorithm that leaves Zuntouched and is guaranteed, under mild assumptions, to converge to astrong stationary solution. While this approach is satisfactory from alocal analysis point of view, it does not settle the main challenge, thatis, aiming for an optimal or near-optimal solution. In our view, progressin this direction will be achieved by addressing problems with specificstructures, such as the BBBP.

Acknowledgments

We would like to thank the following collaborators with whom mostof the results have been obtained: Charles Audet, Luce Brotcorne,

Benot Colson, Pierre Hansen, Brigitte Jaumard, Joachim Judice, Mar-tine Labbe, Sebastien Roch, Frederic Semet and Lus Vicente.

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