Bilevel Programming and the Separation Problem Andrea Lodi - - PowerPoint PPT Presentation

Bilevel Programming and the Separation Problem Andrea Lodi

University of Bologna, Italy andrea.lodi@unibo.it joint work with Ted K. Ralphs and Gerhard J. Woeginger January 9, 2012 @ Aussois

• A. Lodi, Bilevel Programming and the Separation Problem

Context

• We consider a general Mixed Integer Linear Program (MIP) in the form

min{cTx : Ax ≥ b, x ≥ 0, xj integer, j ∈ I} (1) and we do not assume matrix A having any special structure.

• A. Lodi, Bilevel Programming and the Separation Problem

1

Context

• We consider a general Mixed Integer Linear Program (MIP) in the form

min{cTx : Ax ≥ b, x ≥ 0, xj integer, j ∈ I} (1) and we do not assume matrix A having any special structure.

• We are considering a solution method based on branch and bound and bounds computed by

iteratively solving the LP relaxations through a general-purpose LP solver.

• A. Lodi, Bilevel Programming and the Separation Problem

1

Context

• We consider a general Mixed Integer Linear Program (MIP) in the form

min{cTx : Ax ≥ b, x ≥ 0, xj integer, j ∈ I} (1) and we do not assume matrix A having any special structure.

• We are considering a solution method based on branch and bound and bounds computed by

iteratively solving the LP relaxations through a general-purpose LP solver.

• We are interested in one of the major components of MIP technology, namely cutting plane

generation, where the LP relaxation at hand is iteratively strengthened through the addition of valid (linear) inequalities.

• A. Lodi, Bilevel Programming and the Separation Problem

1

Context

• We consider a general Mixed Integer Linear Program (MIP) in the form

min{cTx : Ax ≥ b, x ≥ 0, xj integer, j ∈ I} (1) and we do not assume matrix A having any special structure.

• We are considering a solution method based on branch and bound and bounds computed by

iteratively solving the LP relaxations through a general-purpose LP solver.

• We are interested in one of the major components of MIP technology, namely cutting plane

generation, where the LP relaxation at hand is iteratively strengthened through the addition of valid (linear) inequalities.

• In this talk we discuss the relationship between bilevel programming and cutting plane

generation.

• A. Lodi, Bilevel Programming and the Separation Problem

1

Bilevel Programming

• Multi-level programs are a framework devised by Candler, Norton and Townsley (1977, 1982)

and derive from a problem of agricultural development which was analyzed by the World Bank.

• A. Lodi, Bilevel Programming and the Separation Problem

2

Bilevel Programming

• Multi-level programs are a framework devised by Candler, Norton and Townsley (1977, 1982)

and derive from a problem of agricultural development which was analyzed by the World Bank.

• In that setting, once economic policy makers set certain parameters of agricultural policy,

farmers were viewed as then optimizing their criteria, which differed from that of the policy makers.

• The problem, then, was to set the policy parameters to achieve an optimal effect from the

policy perspective, after understanding how the farmers reacted to these parameters.

• A. Lodi, Bilevel Programming and the Separation Problem

2

Bilevel Programming

• Multi-level programs are a framework devised by Candler, Norton and Townsley (1977, 1982)

and derive from a problem of agricultural development which was analyzed by the World Bank.

• In that setting, once economic policy makers set certain parameters of agricultural policy,

farmers were viewed as then optimizing their criteria, which differed from that of the policy makers.

• The problem, then, was to set the policy parameters to achieve an optimal effect from the

policy perspective, after understanding how the farmers reacted to these parameters.

• Roughly speaking, in bilevel programming one is solving an optimization problem over the set
• f optimal solutions of another optimization problem.
• A. Lodi, Bilevel Programming and the Separation Problem

2

Bilevel Programming

• Multi-level programs are a framework devised by Candler, Norton and Townsley (1977, 1982)

and derive from a problem of agricultural development which was analyzed by the World Bank.

• In that setting, once economic policy makers set certain parameters of agricultural policy,

farmers were viewed as then optimizing their criteria, which differed from that of the policy makers.

• The problem, then, was to set the policy parameters to achieve an optimal effect from the

policy perspective, after understanding how the farmers reacted to these parameters.

• Roughly speaking, in bilevel programming one is solving an optimization problem over the set
• f optimal solutions of another optimization problem.
• This was a bilevel game, but the more general problem of multiple levels was also defined.
• A. Lodi, Bilevel Programming and the Separation Problem

2

The P-hierarchy

• Informally, the P-hierarchy is a scheme for classifying multi-level and multi-stage decision

problems that extends the classes P and NP to problems with multiple decision-makers (and multiple objectives, in the case of optimization models).

• The set of problems on level zero, denoted as Σp

0, are those that can be solved in polynomial

time, the class usually denoted as P.

• A. Lodi, Bilevel Programming and the Separation Problem

3

The P-hierarchy

• Informally, the P-hierarchy is a scheme for classifying multi-level and multi-stage decision

problems that extends the classes P and NP to problems with multiple decision-makers (and multiple objectives, in the case of optimization models).

• The set of problems on level zero, denoted as Σp

0, are those that can be solved in polynomial

time, the class usually denoted as P.

• Roughly speaking, the class of problems on level k ∈ N+, denoted as Σp

k, are those that can be

solved in nondeterministic polynomial time, given an oracle for problems in the class Σp

k−1.

This means that, for example, Σp

1 = NP.

• It is clear that Σp

j ⊆ Σp k for all j, k ∈ N, j ≤ k, but it is unknown whether any of the

inclusions are strict.

• A. Lodi, Bilevel Programming and the Separation Problem

3

The P-hierarchy

• Informally, the P-hierarchy is a scheme for classifying multi-level and multi-stage decision

problems that extends the classes P and NP to problems with multiple decision-makers (and multiple objectives, in the case of optimization models).

• The set of problems on level zero, denoted as Σp

0, are those that can be solved in polynomial

time, the class usually denoted as P.

• Roughly speaking, the class of problems on level k ∈ N+, denoted as Σp

k, are those that can be

solved in nondeterministic polynomial time, given an oracle for problems in the class Σp

k−1.

This means that, for example, Σp

1 = NP.

• It is clear that Σp

j ⊆ Σp k for all j, k ∈ N, j ≤ k, but it is unknown whether any of the

inclusions are strict.

• The hierarchy was first introduced by Stockmeyer (1977), who showed how to generalize the

well-known satisfiability problem to obtain, for every k ∈ N, a class of problems involving the satisfiability of Boolean formulas in a multi-round game that is complete for Σp

k.

• Jeroslow (1985) noted the relationship between decision games and optimization and showed

that k-level discrete optimization problems are Σp

k-hard.

• A. Lodi, Bilevel Programming and the Separation Problem

3

Cutting Plane Generation

• Most of the time in branch-and-cut algorithms we are interested in solving the so-called

separation problem: Definition 1. The separation problem for a polyhedron Q is to determine for a given ˆ x ∈ Rn whether or not ˆ x ∈ Q and if not, to produce an inequality (¯ α, ¯ β) ∈ Rn+1 valid for Q and for which ¯ α⊤ˆ x < ¯ β, where, most of the times, ˆ x is a feasible solution of the continuous relaxation of the MIP, i.e., the relaxation obtained by dropping the integrality requirement on the xj variables, j ∈ I.

• A. Lodi, Bilevel Programming and the Separation Problem

4

Cutting Plane Generation

• Most of the time in branch-and-cut algorithms we are interested in solving the so-called

separation problem: Definition 1. The separation problem for a polyhedron Q is to determine for a given ˆ x ∈ Rn whether or not ˆ x ∈ Q and if not, to produce an inequality (¯ α, ¯ β) ∈ Rn+1 valid for Q and for which ¯ α⊤ˆ x < ¯ β, where, most of the times, ˆ x is a feasible solution of the continuous relaxation of the MIP, i.e., the relaxation obtained by dropping the integrality requirement on the xj variables, j ∈ I.

• Specifically, we will consider the case in which, among all possible (α, β) inequalities that cut
• ff ˆ

x, we want the one maximizing the violation, i.e.: (¯ α, ¯ β) ∈ argmin(α,β)∈Rn+1{α⊤ˆ x − β | α⊤x ≥ β ∀x ∈ Q}. (2)

• A. Lodi, Bilevel Programming and the Separation Problem

4

Cutting Plane Generation (cont.d)

• Because it might be too hard to find a “completely general” inequality in the form α⊤x ≥ β

(cutting off ˆ x), we often restrict ourselves to specific classes (i.e., subsets) of valid inequalities that share some structure.

• A. Lodi, Bilevel Programming and the Separation Problem

5

Cutting Plane Generation (cont.d)

• Because it might be too hard to find a “completely general” inequality in the form α⊤x ≥ β

(cutting off ˆ x), we often restrict ourselves to specific classes (i.e., subsets) of valid inequalities that share some structure.

• Given any of such classes, say C, for a given coefficient vector α ∈ Rn, the calculation of the

right-hand side β required to ensure (α, β) is a member of the class (if such a β exists) may itself be an optimization problem.

• A. Lodi, Bilevel Programming and the Separation Problem

5

Cutting Plane Generation (cont.d)

• Because it might be too hard to find a “completely general” inequality in the form α⊤x ≥ β

(cutting off ˆ x), we often restrict ourselves to specific classes (i.e., subsets) of valid inequalities that share some structure.

• Given any of such classes, say C, for a given coefficient vector α ∈ Rn, the calculation of the

right-hand side β required to ensure (α, β) is a member of the class (if such a β exists) may itself be an optimization problem.

• Two important observations are customary:
• A. Lodi, Bilevel Programming and the Separation Problem

5

Cutting Plane Generation (cont.d)

• Because it might be too hard to find a “completely general” inequality in the form α⊤x ≥ β

(cutting off ˆ x), we often restrict ourselves to specific classes (i.e., subsets) of valid inequalities that share some structure.

• Given any of such classes, say C, for a given coefficient vector α ∈ Rn, the calculation of the

right-hand side β required to ensure (α, β) is a member of the class (if such a β exists) may itself be an optimization problem.

• Two important observations are customary:
• 1. from the one side, in order to minimize (α⊤ˆ

x − β) in (2), a large value of β would be good;

• A. Lodi, Bilevel Programming and the Separation Problem

5

Cutting Plane Generation (cont.d)

• Because it might be too hard to find a “completely general” inequality in the form α⊤x ≥ β

(cutting off ˆ x), we often restrict ourselves to specific classes (i.e., subsets) of valid inequalities that share some structure.

• Given any of such classes, say C, for a given coefficient vector α ∈ Rn, the calculation of the

right-hand side β required to ensure (α, β) is a member of the class (if such a β exists) may itself be an optimization problem.

• Two important observations are customary:
• 1. from the one side, in order to minimize (α⊤ˆ

x − β) in (2), a large value of β would be good;

• 2. on the other hand, given a vector α the value of β must insure validity of the cut, i.e., it

must be not greater than the minimum α⊤x, ∀x ∈ FC, where FC is the closure associated with the class C.

• A. Lodi, Bilevel Programming and the Separation Problem

5

Cutting Plane Generation (cont.d)

• Because it might be too hard to find a “completely general” inequality in the form α⊤x ≥ β

(cutting off ˆ x), we often restrict ourselves to specific classes (i.e., subsets) of valid inequalities that share some structure.

• Given any of such classes, say C, for a given coefficient vector α ∈ Rn, the calculation of the

right-hand side β required to ensure (α, β) is a member of the class (if such a β exists) may itself be an optimization problem.

• Two important observations are customary:
• 1. from the one side, in order to minimize (α⊤ˆ

x − β) in (2), a large value of β would be good;

• 2. on the other hand, given a vector α the value of β must insure validity of the cut, i.e., it

must be not greater than the minimum α⊤x, ∀x ∈ FC, where FC is the closure associated with the class C.

• The above observations lead to formulating the separation problem for maximally violated

inequalities as a bilevel program.

• A. Lodi, Bilevel Programming and the Separation Problem

5

Separation as a Bilevel Program

• More precisely, the separation problem for maximally violated inequalities in the class C with

respect to a given ˆ x ∈ Rn can in principle be formulated mathematically as: min α⊤ˆ x − β (3) α ∈ Cα (4) max β (5) β ≤ α⊤x (6) x ∈ FC, (7) where Cα ⊆ Rn is the set of all vectors that are coefficients for some valid inequality in C.

• A. Lodi, Bilevel Programming and the Separation Problem

6

Separation as a Bilevel Program

• More precisely, the separation problem for maximally violated inequalities in the class C with

respect to a given ˆ x ∈ Rn can in principle be formulated mathematically as: min α⊤ˆ x − β (3) α ∈ Cα (4) max β (5) β ≤ α⊤x (6) x ∈ FC, (7) where Cα ⊆ Rn is the set of all vectors that are coefficients for some valid inequality in C.

• The upper-level objective (3) is to find the maximally violated inequality in the class.

The upper-level constraints (4) require that the inequality is a member of the class.

• A. Lodi, Bilevel Programming and the Separation Problem

6

Separation as a Bilevel Program

• More precisely, the separation problem for maximally violated inequalities in the class C with

respect to a given ˆ x ∈ Rn can in principle be formulated mathematically as: min α⊤ˆ x − β (3) α ∈ Cα (4) max β (5) β ≤ α⊤x (6) x ∈ FC, (7) where Cα ⊆ Rn is the set of all vectors that are coefficients for some valid inequality in C.

• The upper-level objective (3) is to find the maximally violated inequality in the class.

The upper-level constraints (4) require that the inequality is a member of the class.

• The lower-level problem (5)–(7) is to generate the strongest possible right-hand side associated

with a given coefficient vector, i.e., the largest β value among the feasible ones.

• A. Lodi, Bilevel Programming and the Separation Problem

6

Separation as a Bilevel Program

• More precisely, the separation problem for maximally violated inequalities in the class C with

respect to a given ˆ x ∈ Rn can in principle be formulated mathematically as: min α⊤ˆ x − β (3) α ∈ Cα (4) max β (5) β ≤ α⊤x (6) x ∈ FC, (7) where Cα ⊆ Rn is the set of all vectors that are coefficients for some valid inequality in C.

• The upper-level objective (3) is to find the maximally violated inequality in the class.

The upper-level constraints (4) require that the inequality is a member of the class.

• The lower-level problem (5)–(7) is to generate the strongest possible right-hand side associated

with a given coefficient vector, i.e., the largest β value among the feasible ones.

• When does the Bilevel Program (3)–(7) allow a reformulation as a Single Level one?
• A. Lodi, Bilevel Programming and the Separation Problem

6

A good case: Disjunctive cuts

• Given a MIP in the form (1), Balas showed how to derive a valid inequality by exploiting any

disjunction π⊤x ≤ π0

• π⊤x ≥ π0 + 1

(8) where π ∈ ZI × 0C and π0 ∈ Z.

• A. Lodi, Bilevel Programming and the Separation Problem

7

A good case: Disjunctive cuts

• Given a MIP in the form (1), Balas showed how to derive a valid inequality by exploiting any

disjunction π⊤x ≤ π0

• π⊤x ≥ π0 + 1

(8) where π ∈ ZI × 0C and π0 ∈ Z.

• All inequalities valid for the union of the two polyhedra, denoted by P1 and P2 and obtained

by considering the two terms of disjunction (8), are disjunctive cuts.

• A. Lodi, Bilevel Programming and the Separation Problem

7

A good case: Disjunctive cuts

• Given a MIP in the form (1), Balas showed how to derive a valid inequality by exploiting any

disjunction π⊤x ≤ π0

• π⊤x ≥ π0 + 1

(8) where π ∈ ZI × 0C and π0 ∈ Z.

• All inequalities valid for the union of the two polyhedra, denoted by P1 and P2 and obtained

by considering the two terms of disjunction (8), are disjunctive cuts.

• The separation problem can be written as the following bilevel LP:

min α⊤ˆ x − β (9) αj ≥ u⊤Aj − uoπj ∀j (10) αj ≥ v⊤Aj + voπj ∀j (11) u, v, u0, v0 ≥ 0 (12) u0 + v0 = 1 (13) max β (14) β ≤ α⊤x (15) x ∈ P1 ∪ P2. (16)

• A. Lodi, Bilevel Programming and the Separation Problem

7

A good case: Disjunctive cuts (cont.d)

• In other words, one is looking in (15) for a value of β which is:
• A. Lodi, Bilevel Programming and the Separation Problem

8

A good case: Disjunctive cuts (cont.d)

• In other words, one is looking in (15) for a value of β which is:
• 1. feasible with respect to the α vector “decided” in the upper level; and
• 2. as large as possible, i.e., among all β not greater than any α⊤x, ∀x, the largest one is

selected.

• A. Lodi, Bilevel Programming and the Separation Problem

8

A good case: Disjunctive cuts (cont.d)

• In other words, one is looking in (15) for a value of β which is:
• 1. feasible with respect to the α vector “decided” in the upper level; and
• 2. as large as possible, i.e., among all β not greater than any α⊤x, ∀x, the largest one is

selected.

• The feasibility condition x ∈ P1 ∪ P2 is easily guaranteed by any β such that:

β ≤ min{u⊤b − u0π0, v⊤b + v0(π0 + 1)}. (17)

• A. Lodi, Bilevel Programming and the Separation Problem

8

A good case: Disjunctive cuts (cont.d)

• In other words, one is looking in (15) for a value of β which is:
• 1. feasible with respect to the α vector “decided” in the upper level; and
• 2. as large as possible, i.e., among all β not greater than any α⊤x, ∀x, the largest one is

selected.

• The feasibility condition x ∈ P1 ∪ P2 is easily guaranteed by any β such that:

β ≤ min{u⊤b − u0π0, v⊤b + v0(π0 + 1)}. (17)

• However, the “equality” version of (17) above is straightforward to obtain by exploiting the

(negative) sign of β in the objective function (9) and the following easy modeling trick: β ≤ u⊤b − u0π0 (18) β ≤ v⊤b + v0(π0 + 1). (19) Then, (18) and (19) above replace the lower-level program in (9)–(16).

• A. Lodi, Bilevel Programming and the Separation Problem

8

A “bad” case: Capacity Constraints for CVRP

• In the classical Capacitated Vehicle Routing Problem (CVRP) a quantity di of a single

commodity is to be delivered to each customer i ∈ N = {1, . . . , n} from a central depot {0} using a homogeneous fleet of k vehicles, each with capacity K.

• A. Lodi, Bilevel Programming and the Separation Problem

9

A “bad” case: Capacity Constraints for CVRP

• In the classical Capacitated Vehicle Routing Problem (CVRP) a quantity di of a single

commodity is to be delivered to each customer i ∈ N = {1, . . . , n} from a central depot {0} using a homogeneous fleet of k vehicles, each with capacity K.

• Capacity constraints are in the form:
• e={i,j}∈E

i∈S,j∈S

xe ≥ 2b(S) ∀S ⊂ N, |S| > 1 (20) where b(S) is any lower bound on the number of vehicles required to serve customers in set S.

• A. Lodi, Bilevel Programming and the Separation Problem

9

A “bad” case: Capacity Constraints for CVRP

• In the classical Capacitated Vehicle Routing Problem (CVRP) a quantity di of a single

commodity is to be delivered to each customer i ∈ N = {1, . . . , n} from a central depot {0} using a homogeneous fleet of k vehicles, each with capacity K.

• Capacity constraints are in the form:
• e={i,j}∈E

i∈S,j∈S

xe ≥ 2b(S) ∀S ⊂ N, |S| > 1 (20) where b(S) is any lower bound on the number of vehicles required to serve customers in set S. Constraints (20) are a generalization of the TSP subtour elimination constraints and enforce both the connectivity of the solution and that no route has demand exceeding the capacity K.

8 7 1 4 capacity K=10

S

• A. Lodi, Bilevel Programming and the Separation Problem

9

A “bad” case: Capacity Constraints for CVRP (cont.d)

• Independently of the lower bound on the number of vehicles one might want to use, the

separation problem for (20) is inherently bilevel.

• A. Lodi, Bilevel Programming and the Separation Problem

10

A “bad” case: Capacity Constraints for CVRP (cont.d)

• Independently of the lower bound on the number of vehicles one might want to use, the

separation problem for (20) is inherently bilevel. One is looking for a set of customers ¯ S for which the associated inequality (20) is violated and it is useful to define binary variables: ⋆ yi = 1 if customer i belongs to ¯ S, and ⋆ ze = 1 if edge e belongs to δ( ¯ S).

• A. Lodi, Bilevel Programming and the Separation Problem

10

A “bad” case: Capacity Constraints for CVRP (cont.d)

• Independently of the lower bound on the number of vehicles one might want to use, the

separation problem for (20) is inherently bilevel. One is looking for a set of customers ¯ S for which the associated inequality (20) is violated and it is useful to define binary variables: ⋆ yi = 1 if customer i belongs to ¯ S, and ⋆ ze = 1 if edge e belongs to δ( ¯ S).

• Thus, the bilevel separation problem for capacity constraints reads as follows:

min

• e∈E

ˆ xeze − 2b( ¯ S) (21) ze ≥ yi − yj ∀e = {i, j} (22) ze ≥ yj − yi ∀e = {i, j} (23) max b( ¯ S) (24) b( ¯ S) is a valid lower bound.

• A. Lodi, Bilevel Programming and the Separation Problem

10

A “bad” case: Capacity Constraints for CVRP (cont.d)

• Independently of the lower bound on the number of vehicles one might want to use, the

separation problem for (20) is inherently bilevel. One is looking for a set of customers ¯ S for which the associated inequality (20) is violated and it is useful to define binary variables: ⋆ yi = 1 if customer i belongs to ¯ S, and ⋆ ze = 1 if edge e belongs to δ( ¯ S).

• Thus, the bilevel separation problem for capacity constraints reads as follows:

min

• e∈E

ˆ xeze − 2b( ¯ S) (21) ze ≥ yi − yj ∀e = {i, j} (22) ze ≥ yj − yi ∀e = {i, j} (23) max b( ¯ S) (24) b( ¯ S) is a valid lower bound. (25)

• The lower-level program must be specified and the strongest possible inequality corresponds to

solve as lower-level program the 1-dimensional Bin Packing Problem.

• A. Lodi, Bilevel Programming and the Separation Problem

10

A “bad” case: Capacity Constraints for CVRP (cont.d)

• However, there is something intuitively strange here:

– On the one side, one wants to find the largest possible right-hand side, i.e., the largest possible (valid) bound (namely, the exact solution) of the bin packing problem;

• A. Lodi, Bilevel Programming and the Separation Problem

11

A “bad” case: Capacity Constraints for CVRP (cont.d)

• However, there is something intuitively strange here:

– On the one side, one wants to find the largest possible right-hand side, i.e., the largest possible (valid) bound (namely, the exact solution) of the bin packing problem; – On the other hand, this is a minimization problem: find the minimum number of bins!

• A. Lodi, Bilevel Programming and the Separation Problem

11

A “bad” case: Capacity Constraints for CVRP (cont.d)

• However, there is something intuitively strange here:

– On the one side, one wants to find the largest possible right-hand side, i.e., the largest possible (valid) bound (namely, the exact solution) of the bin packing problem; – On the other hand, this is a minimization problem: find the minimum number of bins!

• In other words, there is a conflict between the upper and lower level objective functions that

“do not agree” in sign:

• ne would be pushed to use as many vehicles (bins) as possible to make the capacity constraint

mathematically strong BUT the minimum number of them is required for validity.

• A. Lodi, Bilevel Programming and the Separation Problem

11

A “bad” case: Capacity Constraints for CVRP (cont.d)

• However, there is something intuitively strange here:

– On the one side, one wants to find the largest possible right-hand side, i.e., the largest possible (valid) bound (namely, the exact solution) of the bin packing problem; – On the other hand, this is a minimization problem: find the minimum number of bins!

• In other words, there is a conflict between the upper and lower level objective functions that

“do not agree” in sign:

• ne would be pushed to use as many vehicles (bins) as possible to make the capacity constraint

mathematically strong BUT the minimum number of them is required for validity.

• What seems to be intuitively complex is actually theoretically hard

Theorem 1. The Bilevel Separation Problem for Capacity Constraints is Σp

2-hard.

• A. Lodi, Bilevel Programming and the Separation Problem

11

A “bad” case: Capacity Constraints for CVRP (cont.d)

• However, there is something intuitively strange here:

– On the one side, one wants to find the largest possible right-hand side, i.e., the largest possible (valid) bound (namely, the exact solution) of the bin packing problem; – On the other hand, this is a minimization problem: find the minimum number of bins!

• In other words, there is a conflict between the upper and lower level objective functions that

“do not agree” in sign:

• ne would be pushed to use as many vehicles (bins) as possible to make the capacity constraint

mathematically strong BUT the minimum number of them is required for validity.

• What seems to be intuitively complex is actually theoretically hard

Theorem 1. The Bilevel Separation Problem for Capacity Constraints is Σp

2-hard.

• Proof (sketch): reduction from the 2-QUANTIFIED 1-IN-3-SAT.
• A. Lodi, Bilevel Programming and the Separation Problem

11

Consequences and Conclusions

• It is easy to see that the above complexity result implies that the separation problem for CVRP

Capacity Constraints cannot be reformulated as a single level MIP, unless Σp

2 = NP, i.e., the P

complexity hierarchy collapses.

• Actually, this problem is among the most natural examples of problems that fall in the Σp

2-hard

class.

• A. Lodi, Bilevel Programming and the Separation Problem

12

Consequences and Conclusions

• It is easy to see that the above complexity result implies that the separation problem for CVRP

Capacity Constraints cannot be reformulated as a single level MIP, unless Σp

2 = NP, i.e., the P

complexity hierarchy collapses.

• Actually, this problem is among the most natural examples of problems that fall in the Σp

2-hard

class.

• We have shown an interpretation of the separation problem as Bilevel Programming.
• However, Bilevel Programming is not only an elegant way of expressing separation problems,

but also adds insights on their degree of complexity.

• A. Lodi, Bilevel Programming and the Separation Problem

12

Consequences and Conclusions

• It is easy to see that the above complexity result implies that the separation problem for CVRP

Capacity Constraints cannot be reformulated as a single level MIP, unless Σp

2 = NP, i.e., the P

complexity hierarchy collapses.

• Actually, this problem is among the most natural examples of problems that fall in the Σp

2-hard

class.

• We have shown an interpretation of the separation problem as Bilevel Programming.
• However, Bilevel Programming is not only an elegant way of expressing separation problems,

but also adds insights on their degree of complexity.

• From the theoretical side, further work should be devoted to characterize under which

conditions the reformulation exists (e.g., complexity of the right-hand-side generation, compactness of the associated membership problem, . . . ), although a full characterization is probably as hard of fully characterizing the P-hierarchy itself.

• From a computational viewpoint, directly solving the associated bilevel separation problem,

though difficult, might help in practice as shown by Mattia for metric inequalities.

• A. Lodi, Bilevel Programming and the Separation Problem

12