Most approaches to tackle BQP first of all get rid of the quadratic terms xixj in the objective
function as they are one reason for the high complexity. Note that a quadratic term exists if the corresponding entry in Q is non-zero. We denote the set of all pairs appearing in a quadratic monomial in the objective function with Q := {{i, j} ∈ N × N | Qij + Qji 6= 0},
where N := {1, . . . , n}.
A common way is to replace the monomials by auxiliary variables which yields a linear description in a higher dimension. The first approaches restricted the new variables to be binary, see e. g. [10, 51, 71, 121], which then was extended to approaches with continuous positive variables out of which we particularly consider the classical method of Glover and Woolsey [84], called standard linearization, in the following. Here, for each monomial xixj with {i, j} ∈ Q one
auxiliary variable yij is introduced. Note that by this definition yij and yji denote the same
variable. For the ease of notation we also define qij = Qij+Qji. The new product variables yij
are linked to the original linear variables by adding the following linear inequalities to BQP.
yij ≤ xi, xj ∀ {i, j} ∈ Q (3.1)
yij ≥ xi+ xj− 1 ∀ {i, j} ∈ Q (3.2)
Additionally, the auxiliary variables have to be nonnegative. For the ease of notation we write y ∈ {0, 1}|Q| in the following.
The first set of inequalities, (3.1), makes sure that the product variable obtains a value of zero if one or both of the corresponding linear variables is zero. If both linear variables xi and xj
are set to one, (3.2) leads to yij ≥ 1 and with binarity of y to a value of one. Hence the value
of every yij is exactly the product of xi and xj whenever x ∈ {0, 1}n.
We denote the resulting linearized problem with LBQP, which then is equivalent to BQP in terms of the x-variables and of the objective value. In other words, an optimal solution of LBQP directly leads to an optimal solution of BQP by projecting out the auxiliary y-variables, and, vice versa, each optimal solution can be lifted to an optimal solution of LBQP by setting yij = xixj
for all {i, j} ∈ Q. The dimension of LBQP thus rises by the number of product terms. We denote the convex hull of all integral solutions with
P{0,1}ql := convn(x, y) ∈ {0, 1}n×|Q| Ax ≤ b, (3.1), (3.2)o and the polytope of the relaxed problem with
3.1. LINEARIZATION 27 If the underlying BQP is unconstrained, P{0,1}ql = conv{(x, y) ∈ {0, 1}n×|Q|| (3.1), (3.2)} is called boolean quadric polytope, which was introduced and widely analyzed by Padberg [139]. By reasons of complexity one cannot expect that P{0,1}ql = Pql. Nevertheless the optimal value of Pql defines a lower bound which was shown to be equal to other lower bounds mentioned in the literature [98]. Other linearization approaches are motivated by maintaining the number of auxiliary variables possibly small, such as the compact linearizations of Glover [83], of Adams, Forrester and Glover [2] and of Hansen and Meyer [101]. They achieve the same lower bounds as the standard linearization [101].
To gain better bounds, Adams and Sherali [4] investigated additional possibilities to strenghten the standard linearization (3.1) and (3.2) as much as possible. They proposed to strengthen the linear constraints by multiplying with linear variables, exploiting their binarity. More precisely, in a first step, each inequality Pi∈Naixi ≤ b is multiplied with xj separately for each j ∈ N .
Then, all resulting products xixj are replaced by the corresponding linearization variables yij
if i 6= j, and the product xixi is simply replaced by xi due to the binarity of x. Hence, the
stronger inequalities X
i∈N \{j}
aiyij+ (aj− b)xj ≤ 0 ∀ j ∈ N
are obtained, which even dominate the original ones. The same holds for a multiplication with the complement 1 − xf, separately for each f ∈ E:
X
i∈N \{j}
ai(xi− yij) + bxj ≤ b ∀ j ∈ N.
The same procedure can be applied for all equations of BQP. Here, the multiplication with xj
suffices; a multiplication with 1 − xj yields the same results. Therefore, also the equations
X
i∈N \{j}
aiyij+ (aj− b)xi = 0 ∀ j ∈ N
can be added to LBQP. Due to the authors we call this method Sherali-Adams reformulation or linearization. The advantage of this method are the significantly stronger bounds resulting from the combination of several linearization variables in each inequality. Moreover, the standard linearization is implicitely contained in the Sherali-Adams reformulation when also multiplying the bounds xi ≥ 0 and xi≤ 1 with 1 − xj for all {i, j} ∈ Q.
Applying this procedure d ≤ n times yields the Sherali-Adams reformulation of order d: Let J1, J2 ⊆ N be disjoint subsets with |J1∪ J2| = d, then each original inequality is multiplied
with d many variables or complements, X i∈N aixi− b · Y j∈J1 xj Y j∈J2 (1 − xj) ≤ 0,
and, after a replacement of each product term Πj∈Jxj by a linearization variable yJ, the convex
hull of the feasible solutions is denoted with Pd. Sherali and Adams proved that its projec-
tion PP d := {x ∈ [0, 1]|E| | (x, y) ∈ Pd} yields a better approximation of the original polytope
with increasing d, and, moreover, that the reformulation of order n is exactly the convex hull of all integer points of P . This yields
P ⊇ PP 1 ⊇ PP 2 ⊇ . . . ⊇ PP n= P{0,1},
and an exact linear reformulation of the quadratic problem. By reasons of complexity it follows that the LP size increases exponentially in order d. But even more, the number of additional constraints depends on both the number of linear variables and the number of the original
28 CHAPTER 3. BINARY QUADRATIC OPTIMIZATION PROBLEMS
constraints such that this approach becomes too complex in practice even for small d. Note that even in the case of d = 1 the number of constraints can become extremely high.
Moreover, the multiplication with d linear variables createsPdi=1 nilinearization variables and lifts the corresponding polytope Pd in an extremely high dimension. In the case of d = 1,
only variables of quadratic terms are created, which also is done when applying the standard linearization. But in case that not all quadratic terms appear in the objective function of (BQP), i. e., if |Q| is small, the reformulation technique creates additional quadratic terms respective linearization variables when variables are multiplied where no corresponding quadratic term exists. By this, the problem size is artificially increased and especially in the case with very few or even a single quadratic term this method obviously becomes inefficient.
Another strengthening of the standard linearization was presented by Caprara in [37]. Here, the fact is used that xi = 1 implies yij = xj, such that (yij)j∈N ∈ P{0,1} with yii = xi. With the
definition Pxi {0,1}:= ( {0n} if x i = 0 P{0,1} if xi = 1
and the linear constraints defining P{0,1}, we can write equivalently
Pxi {0,1}= (yij)j∈N ∈ {0, xi}n X j∈N \{i} akjyij+ (aki− bk)xi ≤ 0, ∀ k ∈ {1, . . . , m}
and obtain the linearization (LINsep) min
X i∈N cixi+ X i6=j∈N qijyij s. t. x ∈ P{0,1} (yij)j∈N ∈ P{0,1}xi ∀ i ∈ N yij ≤ xi ∀ i 6= j ∈ N yij ≥ xi+ xj − 1 ∀ i 6= j ∈ N
which is stronger than the standard linearization. By relaxing the latter constraints the problem gets tractable and leads to good dual bounds, see Section 3.3. The disadvantage again is that in the case where originally only few quadratic terms appear in the objective function a large number of additional linearization variables are created which would not be necessary in the standard linearization approach.
In addition there exist several other linearization approaches, which mostly are established for special cases of BQPs, such as the quadratic knapsack problem [20] and variants of the quadratic assignment problem [3, 113, 127]. A survey of linearization approaches can be found, e. g., in [100]. As it fits best for our purpose, we concentrate on the standard linearization and the corresponding polyhedra P{0,1}ql and Pql.