Solving 0â1 Quadratic Programs by Reformulation Techniques

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On Solving 0-1 Quadratic Programs by Reformulation Techniques Ray Pörn, Otto Nissfolk, Anders Skjäl, and Tapio Westerlund Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b01270 • Publication Date (Web): 12 Jul 2017 Downloaded from http://pubs.acs.org on July 18, 2017

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Industrial & Engineering Chemistry Research

On Solving 0-1 Quadratic Programs by Reformulation Techniques Ray P¨orn,∗,† Otto Nissfolk,‡ Anders Skj¨al,‡ and Tapio Westerlund∗,‡ †Faculty of Education and Welfare Studies, ˚ Abo Akademi University, Strandgatan 2, 65100 Vasa, Finland ‡Faculty of Science and Engineering, ˚ Abo Akademi University, Biskopsgatan 8, 20500 ˚ Abo, Finland E-mail: [email protected]; [email protected]

Abstract In this paper we derive and study a reformulation technique for general 0-1 quadratic programs (QP) that uses diagonal as well as non-diagonal perturbation of the objective function. The technique is an extension of the Quadratic Convex Reformulation (QCR) method developed by Billionnet and co-workers, adding non-diagonal perturbations whereas QCR is in a sense diagonal. In this work a set of redundant ReformulationLinearization technique (RLT) inequalities are included in the problem. The redundant inequalities are used to induce non-diagonal perturbations of the objective function that improves the bounding characteristics of the continuous relaxation. The optimal convexification is obtained from the solution of a semidefinite program. We apply the Non-Diagonal QCR (NDQCR) technique to four different types of problems and compare the bounding properties and solution times with the original QCR method. The proposed method outperforms the original QCR method on all four types of test problems.

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Introduction It is well-known that semidefinite programming can be used to derive strong convex relaxations of hard quadratic optimization problems. 1–3 Billionnet et al. 4 introduced another application of semidefinite programming when they reformulated linearly constrained 0-1 QP problem in an optimal way. Their method is called (QCR) and turns non-convex instances into convex ones while keeping the feasible region unaffected. This work was extended in a second paper 5 to include reformulation of mixed integer quadratic programs with convex continuous parts and very recently the same authors presented a compact variant (CQCR) of their technique. 6 Galli and Letchford 7 generalize and simplify the QCR technique to include convexification of 0-1 quadratically constrained quadratic programs (QCQP) and even some cases of mixed integer QCQPs. The reformulation technique proposed by Billionnet et al. 4 transforms nonconvex instances of 0-1 QPs into convex instances with good bounding properties. The same is achieved by Galli and Letchford 7 but in a more general way, by reformulating non-convex 0-1 QCQPs into convex mixed integer QCQPs. This work is positioned in between those two papers. The technique proposed in this paper is to transform a 0-1 QP into a convex mixed integer quadratic program (MIQP). The reformulation is constructed by first adding a set of redundant quadratic constraints to the 0-1 QP and thereafter solving a semidefinite relaxation of the strengthened 0-1 QP. Finally, the dual variables of the quadratic constraints are used to convexify the objective function and additional continuous variables are included to linearize some non-convex quadratic parts. This convexification leads to a bound at least as good as the bound obtained from the QCR method. In practice, the obtained bound is always better. During the preparation of this paper we became aware of the fact that Ji et al. 8 used a similar idea in the context of quadratic Knapsack problems. Our derivation differs from theirs but the basic ideas of the techniques coincide. In this paper we apply the reformulation technique to a broader spectrum of general linearly constrained 0-1 QP problems, whereas Ji et al. only considered the specific case of knapsack problems. Ji


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et al. did not suggest a name for the reformulation, throughout this article we use the term NDQCR, with ND standing for Non-Diagonal. The paper is organized as follows. In Section 2 we present our reformulation technique and prove the main theoretical results. In Section 3 the reformulation technique is applied to four different classes of problems. First we study reformulation of Boolean least squares problems. Then we describe and study the Coulomb glass problem in the study of lightly doped semiconductors; 9,10 this problem is similar to the more well-known spin glass problem. 11 The third set of problems is task allocations with communication costs. The last set of problems is the taixx c instances from the QAPLIB. 12 These quadratic assignment problems have special structure and they can be converted into 0-1 QPs with only n variables compared to the corresponding QAP formulation with n2 variables. Section 4 concludes the paper. n and Throughout the paper the non-negative orthant of Rn is denoted by R+

the set of matrices with dimension m × n is Rm×n . The set of symmetric matrices of dimension n × n is S n and the standard inner product for matrices is A•B Pn Pn = Tr(AB T ) = j=1 Aij Bij . The notation A 0 means that the matrix A is i=1 positive semidefinite. The diagonal matrix with vector a ∈ Rn along the diagonal is Diag(a) and for any square matrix A∈ Rn×n the diagonal is extracted by diag(A) ∈ Rn . The optimal value of problem (P) is denoted by v(P) and the continuous relaxation of problem P is P. Occasionally the term convex is used for a quadratic problem with a convex continuous relaxation.



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Reformulation of 0-1 QPs A general 0-1 QP with linear constraints is given by min

xT Qx + q T x

s.t. Ax = a (QP01) Bx ≤ b x ∈ {0, 1}n where Q ∈ S n , A ∈ Rm×n , B ∈ Rk×n , q ∈ Rn , a ∈ Rm and b ∈ Rk . The continuous relaxation QP01 is obtained by replacing the binary conditions in QP01 by simple bounds x ∈ [0, 1]n . Problem QP01 can be strengthened by adding various redundant quadratic constraints to the problem. 13–15 In this paper we study the inclusion of squared norm constraints and simple RLT inequalities 16 in particular. These quadratic constraints are given by

kAx − ak2 = xT AT Ax − 2aT |{z} Ax +aT a = xT AT Ax − aT a = 0 a

and the McCormick envelopes 17,18

xi xj ≤ xi , xi xj ≤ xj , xi xj ≥ xi + xj − 1, xi xj ≥ 0 ∀ i 6= j.


(RLT)

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The strengthening procedure turns problem QP01 into a quadratically constrained quadratic program: min

xT Qx + q T x

s.t. Ax = a Bx ≤ b xT AT Ax = aT a xi xj ≥ 0,

(QP01s)

xi xj ≥ xi + xj − 1 ∀ i 6= j

xi xj ≤ xi ,

xi xj ≤ xj

∀ i 6= j

x ∈ {0, 1}n . The continuous relaxation of this problem, QP01s, is evidently non-convex. The standard semidefinite relaxation of this strengthened problem is min

Q • X + qT x

s.t. Ax = a Bx ≤ b diag(X) = x AT A • X = a T a (SDPr) Xij ≥ 0,

Xij ≥ xi + xj − 1 ∀ i 6= j

Xij ≤ xi , Xij ≤ xj   T 1 x  0  x X

∀ i 6= j

x ∈ [0, 1]n , X ∈ S n . Problem SDPr has a bounded feasible region but it does not necessarily satisfy a Slater condition. 7 An alternative way of deriving the semidefinite relaxation is through the Lagrangian relaxation of problem QP01s. When the condition that xi is binary



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is replaced by the quadratic equality constraint x2i = xi the Lagrangian function of QP01s becomes f (x, λ, µ, δ, α, S, T, U, V ) = xT Qx + q T x + λT (Ax − a) + µT (Bx − b) +

n X

δi (x2i − xi )

i=1

+ α(xT AT Ax − aT a) −

n X n X i=1

+

n X n X i=1

j=1

i6=j

Uij (xi xj − xi ) +

Sij xi xj −

j=1

i6=j n n XX i=1

n X n X i=1

Tij (xi xj − xi − xj + 1)+

j=1

i6=j

Vij (xi xj − xj ),

j=1

i6=j

where λ, µ, δ, α, S, T, U, V are multipliers corresponding to the different constraints. k are nonnegative and the symmetric matrix multipliers The multipliers µ ∈ R+

S, T, U, V ∈ S n are elementwise nonnegative with zero diagonal. The multipliers λ ∈ Rm , δ ∈ Rn , α ∈ R are unrestricted in sign. It follows from a result by Faye and Roupin, 13 Proposition 5, that this formulation with a scalar multiplier α is as tight as the larger formulation with a m × n matrix of multipliers that correspond to the whole set of strengthened equality constraints xj Ax = axj , j = 1, . . . , n, as long as there is no duality gap in the SDP solution. After some rearrangement the Lagrangian function can be represented as

ˆ + qˆT x + cˆ f (x, λ, µ, δ, α, S, T, U, V ) = xT Qx

with ˆ = Q(δ, ˆ α, S, T, U, V ) = Q + Diag(δ) + αAT A − S − T + U + V Q qˆ = qˆ(λ, µ, δ, T, U, V ) = q + AT λ + B T µ − δ + p(T, U, V ) cˆ = cˆ(λ, µ, α, T ) = −λT a − µT b − αaT a −

n X n X i=1

j=1

i6=j


Tij

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and the column vector p(T, U, V ) ∈ Rn is defined elementwise as

pi = 2

n X

Tij −

n X

j=1

j=1

i6=j

i6=j

(Uij + Vij ).

The Lagrangian dual is then

ˆ + qˆT x + cˆ inf xT Qx

sup k ,δ∈Rn ,α∈R λ∈Rm ,µ∈R+

x∈Rn

S≥0,T ≥0,U ≥0,V ≥0

which can be represented explicitly as the semidefinite program 2,3 max

t  −t + cˆ s.t.  1 ˆ 2q S ≥ 0,



1 T ˆ  2q

ˆ Q

0

T ≥ 0,

U ≥ 0,

V ≥0 (SDPd)

diag(S) = 0,

diag(T ) = 0,

diag(U ) = 0,

diag(V ) = 0

S, T, U, V ∈ S n λ ∈ Rm ,

k µ ∈ R+ ,

δ ∈ Rn

α, t ∈ R under the assumption that supremum is attained. According to Galli and Letchford 7 the supremum is almost always attainable in practice, this is also our experience. Under this assumption there exist feasible Lagrangian multipliers such that v(SDPr)=v(SDPd). A useful interpretation of the dual solution is that it corresponds to the multipliers that maximize the minimum of the quadratic function over the feasible region of problem QP01. This means that the lower bound of the continuous relaxation is maximal.



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Using the optimal set of multipliers (λ∗ , µ∗ , δ ∗ , α∗ , S ∗ , T ∗ , U ∗ , V ∗ ) we denote ˆ ∗ = Q(δ ˆ ∗ , α∗ , S ∗ , T ∗ , U ∗ , V ∗ ), Q qˆ∗ = qˆ(λ∗ , µ∗ , δ ∗ , α∗ , S ∗ , T ∗ , U ∗ , V ∗ ), cˆ∗ = cˆ(λ∗ , µ∗ , α∗ , T ∗ ). Using this notation and strong duality the optimal value of the dual problem is equivalent to the optimal value of the unconstrained convex program

ˆ ∗ x + qˆ∗T + cˆ∗ . min xT Q

x∈Rn

In the next step we rewrite problem QP01 using the optimal multipliers from problem SDPr. For this purpose we use the notation ¯ ∗ = Q + Diag(δ ∗ ) + α∗ AT A − S ∗ − T ∗ + U ∗ + V ∗ Q q¯∗ = q + AT λ∗ + B T µ∗ − δ ∗ c¯∗ = −λ∗T a − µ∗T b − α∗ aT a ˆ∗ = Q ¯ ∗ , qˆ∗ = q¯∗ + p(T ∗ , U ∗ , V ∗ ) and cˆ∗ = c¯∗ − and observe that Q

Pn Pn i=1

j=1

i6=j

Tij∗ . Using

this notation a reformulated 0-1 QP problem is constructed where all non-diagonal perturbations are explicitly compensated using sums of bilinear terms. The reformulated problem is

min

∗

xT Q x + q ∗T x + c∗ + 2

n X n X

∗ (Sij + Tij∗ )xi xj

i=1 j=i+1

−2

n X

n X

(Uij∗ + Vij∗ )xi xj

i=1 j=i+1

(QP01rf)

s.t. Ax = a Bx ≤ b x ∈ {0, 1}n .


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∗

The quadratic part xT Q x + q ∗T x + c∗ of the objective is now convex by construction, ¯ ∗ that is guaranteed by the first constraint in due to the positive semi-definiteness of Q problem SDPd together with the property of the Schur complement. The objective as a whole is still non-convex due to the sums that include the bilinear term xi xj . This non-convexity can be removed by a standard linearization of the positive terms and the negative terms separately. The RLT constraints are hereby reintroduced in the problem. Each positive bilinear term is replaced by its corresponding convex envelope and each negative term by its concave envelope. Continuous variables yij and zij (i < j) coupled with constraints

yij ≥ 0,

yij ≥ xi + xj − 1,

zij ≤ xi ,

zij ≤ xj

are included in the problem. We observe that these constraints correspond exactly to the constraints in the semidefinite relaxation that were used to compute the optimal dual variables (non-diagonal perturbations) S ∗ , T ∗ , U ∗ and V ∗ . This linearization procedure gives rise to the following convex mixed integer reformulated problem

min

∗

xT Q x + q ∗T x + c∗ + 2

−2

n X n X i=1 j=i+1 n X n X

∗ (Sij + Tij∗ )yij

(Uij∗ + Vij∗ )zij

i=1 j=i+1

s.t. Ax = a (MIQP)

Bx ≤ b yij ≥ 0,

yij ≥ xi + xj − 1

zij ≤ xi ,

zij ≤ xj

x ∈ {0, 1}n yij , zij ∈ R+ (i < j). This is a mixed integer quadratic program with a convex objective function, n binary variables, n2 − n continuous variables and 2n2 − 2n additional linear inequality



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constraints. Theorem 1. Problems QP01, QP01rf and MIQP have the same optimal value, i.e. v (QP01) = v (QP01rf) = v(MIQP). Proof. The feasible region of QP01 and QP01rf are identical. Given any feasible binary vector x ∈ {0, 1}n we have ∗

xT Q x + 2

n X n X

∗ (Sij + Tij∗ )xi xj − 2

i=1 j=i+1

n X n X

(Uij∗ + Vij∗ )xi xj + q ∗ T x + c∗

i=1 j=i+1

= xT (Q + Diag (δ ∗ ) + α∗ AT A − S ∗ − T ∗ + U ∗ + V ∗ )x + 2

n X n X


i=1 j=i+1

−2

n X n X

(Uij∗ + Vij∗ )xi xj + (q − δ ∗ )T x − α∗ aT a

i=1 j=i+1

= xT Qx+q T x −xT (S ∗ + T ∗ ) x + 2

n X n X


i=1 j=i+1

|

+

0 due to symmetry of n X ∗ ∗

xT (U + V )x − 2

S∗

{z

}

and T ∗ and zeros on diagonals

n X

(Uij∗ + Vij∗ )xi xj

i=1 j=i+1

{z

|

}

0 due to symmetry of U ∗ and V ∗ and zeros on diagonals

+ Pn

xT Diag(δ ∗ )x − δ ∗ T x | P {z }

∗ 2 i=1 δi xi −

n ∗ i=1 δi xi =0

due to binary x

∗ T T ∗ T T T +α | x A Ax {z − α a a} = x Qx+q x. 0 due to Ax=a

For any feasible binary vector x the objective values of QP01 and QP01rf are identical. Thus, the two problems are equivalent and their optimal values coincide. Due to the minimization and signs of matrix multipliers in MIQP at least one of the constraints yij ≥ 0, y ij ≥ xi + xj − 1 will be active at optimum. The same holds for zij ≤ xi and zij ≤ xj . Then yij = max {0, xi + xj − 1} = xi xj and zij = min {xi , xj } = xi xj since xi and xj are binary. At optimum we can replace every occurrence of yij and zij by the product xi xj and problem QP01rf is obtained so it holds that v (QP01) = v (QP01rf) = v(MIQP). Theorem 2. Assume strong duality, v (SDPr) = v(SDPd), and that there exists a


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set of optimal multipliers. The optimal value of problem MIQP is then equal to the optimal value of the semidefinite relaxation SDPr and its dual SDPd, i.e. v MIQP = v (SDPr) = v(SDPd). Proof. By construction the multipliers (λ∗ , µ∗ , δ ∗ , α∗ , S ∗ , T ∗ , U ∗ , V ∗ ) are optimal ˆ ∗ x + qˆ∗T x +ˆ in SDPd and the optimal value of SDPd is equal to minx∈Rn xT Q c∗ . Due ˆ ∗ < 0. This problem is to the existence of a dual optimal solution it holds that Q thereby convex. From primal feasibility in SDPr it follows that x ∈ [0, 1]n and since the linear constraints Ax = a and Bx ≤ b are incorporated in the objective function with optimal multipliers λ∗ and µ∗ the problem above is equivalent to the linearly constrained convex problem ˆ ∗ x + qˆ∗T x + cˆ∗ xT Q

min

s.t. Ax = a Bx ≤ b x ∈ [0, 1]n . ∗

The objective function is now rewritten using notations Q , q ∗ and c∗ . ∗

ˆ ∗ x + qˆ∗T x + cˆ∗ = xT Q x + (q ∗ + p∗ )T x + c∗ − xT Q

n X n X i=1

Tij∗

j=1

i6=j T

∗

=x Q x+q

∗T

∗

x+c +p

∗T

x−

n X n X i=1

Tij∗

j=1

i6=j

| (?) = p∗T x −

n X n X i=1

Tij∗ =

{z

n X

p∗i xi −

n X n X

i=1

j=1

i=1

i6=j

j=1



n n n n X n X X X  X  ∗ ∗ ∗  2 T − U + V x − Tij∗ ij ij ij  i  i=1

j=1

n X n X i=1

j=1

i6=j

∗ Sij 0

i=1

j=1

i6=j

=

Tij∗

i6=j

 =

}

(?)

+

i6=j n X n X i=1

Tij∗

(xi + xj − 1) −

j=1

i6=j n X n X i=1

j=1

j=1

i6=j

i6=j


Uij∗ xi

−

n n X X i=1

j=1

i6=j

Vij∗ xj


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The last equality comes from regrouping and symmetry of the matrix multipliers. A ∗ is also included. We conclude that zero sum including multipliers Sij

∗

ˆ ∗ x + qˆ∗T x + cˆ∗ = xT Q x + q ∗ T x + c∗ + (?) . xT Q

Continuous variables yij and zij are now included along with the constraints yij ≥ max{0, xi + xj − 1} and zij ≤ min{xi , xj } and incorporated in the problem as

min

∗

xT Q x + q ∗T x + c∗ +

n X n n X n X X ∗ (Sij + Tij∗ )yij − (Uij∗ + Vij∗ )zij i=1

j=1

i6=j

i=1

j=1

i6=j

s.t. Ax = a Bx ≤ b yij ≥ max{0, xi + xj − 1} zij ≤ min{xi , xj } n×n x ∈ [0, 1]n , y, z ∈ R+ .

This problem is now precisely MIQP. Due to minimization the RLT constraints will be active at optimum so yij = max{0, xi + xj − 1} and zij = min{xi , xj } and the optimal value of the problem above will coincide with min

ˆ ∗ x + qˆ∗T x + cˆ∗ xT Q

s.t. Ax = a Bx ≤ b x ∈ [0, 1]n , which has a minimal value equal to v(SDPd). Now it holds that v (SDPd) = v (SDPr) = v(MIQP) as claimed. Theorem 2 guarantees that the bounding properties of MIQP are good when the problem is solved using a branch and bound framework. At the root node the continuous relaxation of MIQP is as good as the semidefinite relaxation of the strengthened


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problem QP01s. The original QCR method uses only diagonal perturbations (δ) and constraint induced perturbations (α). Since these perturbations are a subset of those used in this paper it is clear that the relaxation obtained by the QCR method is looser than the bound obtained from solving problem MIQP. The reformulation technique is summarized in the following steps.

Non-diagonal quadratic convex reformulation technique (NDQCR) Given a general QP01 problem. 1. Strengthen the problem by including a set of RLT inequalities and squared norm constraints. 2. Solve the semidefinite relaxation (SDPr) and its dual (SDPd). 3. Collect the multiplier values and form problem MIQP. 4. Solve problem MIQP using any suitable solver. Remark 1 Since multipliers λ and µ do not participate in the quadratic part they can be put to zero. This has no effect on the quality of the lower bound. Remark 2 Due to computational aspects it is not always possible to include all RLT inequalities in the semidefinite relaxation. The technique is also valid if only a subset of inequalities is included. This comes, of course, at some expense of the quality of the bound. This is a common situation in practice. Remark 3 Variables yij and zij that correspond to zero or small multiplier values may be omitted, since their impact on the quality of the bounding is small. This may sometimes reduce the size of the convexified MIQP problem significantly. We end this paragraph by a simple illustration of the proposed method. Example 1 Consider the QP problem 4 min

xT Qx + q T x

s.t. Ax = a x ∈ {0, 1}5



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with 











−24 2 18 −12   0  −9   1         −24  −7   1  0 −3.5 18 −42                    Q= 2  , q =  2  , AT =  0  , a = 2. −3.5 0 20 2              18     1  18 20 0 −44 23             −12 −42 2 −44 0 12 1 The optimal solution is x∗ = (0, 1, 1, 0, 1) with optimal value −80. This problem is reformulated in five different ways using different perturbation strategies: i) α and δ perturbations (QCR method) ii) α, δ and S perturbations iii) α, δ and T perturbations iv) α, δ and U perturbations v) α, δ and V perturbations The results of all relaxations are given below.

Strategy

i)

ii)

iii)

iv)

v)

v(∗)

-88.02

-80

-82.23

-82.20

-83.84

Note that all non-diagonal RLT strategies are strictly better than only diagonal perturbation combined with the squared-norm constraint. Strategy ii) gives the optimal binary solution so the relaxation is sharp. The optimal multipliers for strategy ii) are 







1.99 1.40 56.96 12.66   −15.89   0      4.78   1.99 0 0 32.40 0              ∗ ∗ , δ =  1.00  , α∗ = 113.32. S =  1.40 0 0 22.38 0           −18.07   56.96 32.40 22.38 0 6.36          −25.22 12.66 0 0 6.36 0


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Given these optimal multiplier values problem MIQP is constructed using 

 97.44 87.33

     ∗  ∗ ∗ T ∗ Q = Q + Diag (δ ) + α A A − S =      

87.33

118.10

0.60

−3.50

74.36

98.92

88.66

71.32



0.60

74.36

88.66   −3.50 98.92 71.32     1.00 −2.38 2.00    −2.38 95.26 62.96    2.00 62.96 88.10



 6.89     −11.78        ∗ ∗ q = q − δ =  1.00       41.07      37.22 c∗ = −α∗ aT a = −453.29. ∗

The eigenvalues of Q are (0.50, 2.19, 7.70, 46.24, 343.26) so the objective is convex as expected. The reformulated problem is then min

∗

xT Q x + q ∗ T x + c∗ + 2

X

∗ Sij yij

(i,j)∈I

s.t. x1 + x2 + x4 + x5 = 2 yij ≥ 0, yij ≥ xi + xj − 1

∀ (i, j) ∈ I

x ∈ {0, 1}5 , where the index set I = {(1, 2) , (1, 3) , (1, 4) , (1, 5) , (2, 4) , (3, 4) , (4, 5)} corresponds to non-zero multipliers in S ∗ . This is a convex MIQP problem with 5 binary variables, 7 continuous variables and 15 linear constraints. The optimal value of the continuous relaxation of this convex quadratic problem is −80, i.e. equal to the optimal value of the corresponding semidefinite relaxation.



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Applications In this section the NDQCR technique is applied to four different sets of test problems. The first set is Boolean least squares problems (BLS). These are unconstrained 0-1 QPs with an objective function that is already convex. The second set consists of the so called Coulomb glass problems that are derived from a physical context concerning energy minimization of electron configurations in semiconductor materials. 9,10,14 The third set is task allocation problems with communication costs, using instances similar to those tested by Billionnet et al. 4 The fourth set is a special type of quadratic assignment problems (QAP) that can be reduced to 0-1 QPs with a significantly smaller dimension than the original QAP formulation. The problem models gray scale pattern problems and are labeled as the taixxc instances in the QAPLIB. 12,19–21 The numerical testing was conducted using MATLAB R2013b together with the CVX toolbox 22 and calling the SDPT3 solver 23 for the SDP problems. The convexified problem MIQP were solved with CPLEX 12.2.0.0. The computer used in the testing was an Intel i7 930 @ 2.8GHz with 6 GB RAM running Windows 7 (64-bit).

Boolean least squares BLS is a basic problem in digital communication. The objective is to identify a digital signal x ∈ {0, 1}n from a collection of noisy measurements. The problem can be modeled as min

kDx − dk2

s.t. x ∈ {0, 1}n , where D ∈ Rm×n and d ∈ Rm are given. The continuous relaxation of this unconstrained 0-1 QP is itself a convex program. The reformulation gives a tighter relaxation and improves the gap at the root node. The standard SDP relaxation of this problem is


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min

DT D • X − 2dT Dx + dT d

s.t. diag (X) = x   T  1 x   

Solving 0â1 Quadratic Programs by Reformulation Techniques

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