Introduction on Disjunctive Cutting Planes

y

− ^b_B^⊥. (4.22)

for non-basic constraints.

Note, that some constraints from the set B^⊥ might also be active in addition to the basic constraints indexed by B.

The linear system (4.19) possesses m^ equations in total and contains n original vari-ables(x, y)and m^ slack variabless, withn <m. The solution^ (¯x,y,¯ s¯_B^⊥,¯s_B)of (4.19) is a basic solution for the basis ˜B, with basic variables(¯x,y,¯ ¯s_B^⊥) and non-basic vari-ables s_B satisfying ¯s_B = 0. The matrix representing the linear system (4.19) for all variables (x, y, s_B^⊥, s_B) is given by

A˜ := A^ −I

=

A^B 0 −I A^_B^⊥ −I 0

. (4.23)

Therefore the basis matrix for the basis ˜B according to Definition 4.1 is given by A˜˜B=

A^_B 0 A^_B^⊥ −I

, (4.24)

which is invertible, since A^_B is invertible.

For simplification of subsequent calculations in this chapter as well as in Chapter 5, we introduce the standard notation

a¯kj := −(e^T_kA^⁻¹_B )j,

¯

a_ij := −(^a^T_iA^⁻¹_B )_j, a¯_k0 := (e^T_kA^⁻¹_B ^b_B),

¯

a_i0 := (^a^T_iA^⁻¹_B ^b_B− ^b_i).

(4.25)

4.2 Introduction on Disjunctive Cutting Planes

In this section we introduce some theory for disjunctive cutting planes and the closely related disjunctive programs. It was developed since the early seventies by Balas [10]

and others. Latest results by Perregaard [86] and Balas and Bonami [18], turn these cutting planes into very powerful cut generators for mixed-integer linear solvers. Our

aim is to extend the results, such that disjunctive cutting planes can be applied efficiently for mixed-integer quadratic programs, see Chapter 5.

Reviewing the results of Perregaard [86], we consider a mixed-integer linear program x ∈R^nc, y∈Nⁿⁱ : variables andm denotes the number of constraints. Note, that the constraints contain lower and upper bounds for each variable, see also Section 4.1. The optimal solution of (4.26) is given by(x^∗, y^∗)∈R^nc×Nⁿⁱ. The corresponding continuous relaxation of MILP (4.26) is given by LP (4.12).

Let(¯x,y)¯ ∈R^nc+ni be the solution of the continuous relaxation (4.12). Moreover, let thek-th component of ¯ybe fractional, i.e., ¯y_k∈/ N. As a consequence, eithery^∗_k≥ d¯y_ke ory^∗_k ≤ b¯y_kcholds, since(x^∗, y^∗)∈R^nc×Nⁿⁱ is the optimal solution of problem (4.26) withy^∗_kintegral. This property can be expressed via a logical condition, which is called a disjunction∨, i.e.,

y^∗_k≥ d¯y_ke ∨ y^∗_k≤ b¯y_kc. (4.27) The feasible region of the continuous relaxation (4.12) is truncated by integrating the disjunctive condition (4.27). The resulting relaxation of MILP (4.26) is called disjunctive relaxation and is given by

x ∈R^nc, y∈Rⁿⁱ : is called disjunctive condition. The solution of the disjunctive relaxation (4.28) is denoted by (¯x_DJ,y¯_DJ) ∈ R^nc+ni. Since (¯x,y)¯ is not feasible for the disjunctive relax-ation (4.28) and LP (4.12) is a relaxrelax-ation of (4.28),

c^T

holds. The feasible region of the disjunctive relaxation (4.28) is the union of the two

An optimization problem possessing a linear objective and a feasible region given by the union of finitely many polyhedra, is called a disjunctive program. We denote the finite index set of the corresponding polyhedra byQ, i.e.,P^h, h∈Qis the correspond-ing h-th polyhedron. For the disjunctive relaxation (4.28) on variabley_k ∈N,|Q|=2 and Q={1, 2} with P¹=P_lower^k and P²=P_upper^k holds.

The disjunctive relaxation (4.28) can be used to generate cutting planes for the mixed-integer problem (4.26). For the construction of cutting planes, the existence of a com-pact representation of the convex hull of the union of polyhedra in a higher dimension is exploited. The associated convex hull can be projected on the original space yield-ing so-called disjunctive cuttyield-ing planes. We introduce some related definitions and theorems.

The following theorem of Balas [11] explains the correlation between disjunctive cut-ting planes and the disjunctive relaxation (4.28).

Theorem 4.1. Let Q^∗ 6=∅ be a index set of non-empty polyhedra P^h, h∈ Q, where

The inequality a^T_c

Proof. The proof is closely related to Farkas Lemma, see Balas [11].

The disjunctive relaxation (4.28) is called two-term disjunction, since |Q|= 2 holds.

It is obtained from the general disjunctive program (4.37) by setting

D¹ = −e^T_k, (4.39)

D² = e^T_k, (4.40)

d¹₀ = −b¯y_kc, (4.41)

d²₀ = d¯y_ke, (4.42)

where e_k ∈ Rⁿ denotes the k-th unit vector. Note, that P_lower^k is the polyhedron described by inequalities

while P_upper^k is given by

Disjunctive cutting planes are derived from the disjunctive relaxation (4.28) or more generally (4.37). Theorem 4.1 shows, that any cutting plane a^T_c

x y

≥b_c induced from a disjunctive relaxation has to satisfy conditions (4.38), in order to be valid for the relaxation (4.37).

Therefore, a cutting plane is given by an inequality, which is violated by the solution (¯x,y)¯ of the continuous relaxation (4.12) and which satisfies the conditions of Theo-rem 4.1. As a consequence, a cutting plane can be obtained by solving the so-called cut generating linear program (CGLP_k) corresponding to disjunction (4.27), see e.g., Balas [11]. Note, that the subsequent formulation of the CGLPk, also contains the lower bounds on the variables by using matrix A^ and right hand side b^ instead of A and b, see LP formulations (4.12) and (4.15).

(ac, bc)∈Rⁿ×R, (u, u0)∈R^{m^} ×R, (v, v0)∈R^{m^} ×R:

is called normalization constraint. Note, that(u, u₀)correspond toP^k_lower, while(v, v₀) are associated with P_upper^k , i.e., (u¹, u¹₀) and (u², u²₀) in Theorem 4.1, respectively.

The linear program (4.45) constructs the cutting plane a^T_c x

y

≥b_c, that is most violated at the solution (¯x,y)¯ of the continuous relaxation (4.12), i.e., it minimizes the objective function of the CGLP_k. The constraints of problem (4.45) restrict the choice to those cutting planes that are valid for the union of the polyhedraP^k_lower and P^k_upper defined by (4.31) and (4.32), respectively. The optimization variables (ac, bc) correspond to the coefficients of the resulting cutting plane, whereas the variables (u, u₀, v, v₀)ensure the validity of the constructed cut according to Theorem 4.1.

The cut generating linear program (4.45) obviously depends on the disjunction that is considered. In addition it depends on the point to be cut off and therefore forms the objective function, here (¯x,y). Furthermore, it depends on the constraints of¯ underlying disjunctive program, here determined by the matrixA^ and the right hand sideb. In case we consider different cut generating linear programs, we use the notation^ CGLPk(¯x,y,¯ A,^ b)^ in order to allow a precise distinction.

CGLP_k only considers the integrality condition for variable y_k. Therefore, the gen-erated cutting plane can be strengthened by a procedure of Balas and Jeroslow [14], that takes the integrality conditions for additional integer variables y_i ∈ I\{k} into account. Solving CGLP_k yields one cutting plane. There exists various suggestions how to generate more than just one cutting plane from the disjunctive relaxation corresponding to disjunction (4.27), see Perregaard [86] for details. Furthermore, it is possible to neglect those variablesx_i, or y_i respectively, that take the value of one of the corresponding box-constraints in the solution (¯x,y)¯ of LP (4.12), i.e., ¯x_i = (x_u)_i or ¯x_i = (x_l)_i or ¯y_i = (y_u)_i or ¯y_i = (y_l)_i respectively. As a consequence, the dis-junctive relaxation (4.28) is restricted to a lower dimensional subspace, such that the dimension of CGLP_k is significantly reduced. The cutting plane determined by the corresponding cut generating linear program can be lifted to be valid in the full space, see Perregaard [86].

Further theory on disjunctive programming and related cutting planes can be found in Balas [9, 10, 11, 12], Balas, Ceria, and Cornuejols [13], Balas and Perregaard [15, 16, 17], Ceria and Pataki [37], Ceria and Soares [38] and others.