Eigenvalue related cuts

The possibility to construct additional constraints based on the Gilmore-Lawler bound procedure suggests the use of another well-known bounding technique, that is the eigenvalue based approach by Finke, Burkard, and Rendl [34].

We follow the notation in [34] and denote byxv, wy_`andxv, wy_´ ordered dot products of real vectorsv, wP Rⁿ:

xv, wy_`:“ xv^Ó, w^Óy “ max

XPΠⁿxv,Xwy, xv, wy_´ :“ xv^Ó, w^Òy “ min

XPΠⁿxv,Xwy, (6.10) wherew^Ó andw^Ò denote the vectors towwhose elements are rearranged in non-ascending and non-descending order, respectively. The eigenvalue bound (EVB) is based on the

fact that

@X PΠⁿ: @

λpAq, λpBqD

´ ď@

A, XBX^TD ď@

λpAq, λpBqD

`, (6.11) see [34, Theorem 3].

For the following discussion about eigenvalue related cuts, assume the eigenvalues of A “ řn

i“1µ_ip_ip^T_i to be sorted in non-ascending order, and in non-descending order denote by λ₁ ď λ₂ ď. . . ď λ_n the eigenvalues of B. In [27, Chapter 2.2.2], Ding and Wolkowicz proposed a smart implementation for incorporating EVB into their matrix lifting based relaxation framework. They strengthened their relaxation by applying the cuts

0ď ÿl i“1

xpi,Ypiy ´λ_i for lP t1, . . . , n´1u. (6.12) From the proof of [27, Lemma 2.1], it is clear that (6.12) describes a sensible integration of EVB based conditions.

The incorporation into the respective SDP relaxations is straightforward. However, this does not mean that the presented procedure is similarly reasonable for all regarded relaxations frameworks. To illustrate this circumstance, consider the following result.

Lemma 6.2. Let the QAP instancepA, B, Cqbe given and assume that the approximation tolerance ε is zero. For any feasible point pX,F`,F´,U1, . . . ,Uk,Yq to problem (5.11), the majorization relation

λpYqăλpBq (6.13)

holds valid.

Proof. Let λ_‹ :“ rλ_1‹, . . . , λ_k‹s^T denote the vector consisting of the distinct eigenvalues tλ_i‹u of B, and let tw₁, . . . , w_nu be a set of orthonormal eigenvectors of Y, such that Yw_i “ λ_ipYqw_i for 1 ď i ď n. Furthermore, define the nˆk matrix ˇS :“ pˇs_ijq with elements ˇs_ij “ xw_i,Ujw_iy. Then,

@iP t1, . . . , nu: xw_i,Yw_iy “

k

ÿ

i“j

λ_j‹xw_iUjw_iy “

k

ÿ

j“1

λ_i‹ˇs_ij

reveals the identity λpYq “ Sλˇ _‹.

The equality constraints in (5.11d) and (5.11f) imply

Moreover, due to the positive semidefiniteness of the variables tUju, it follows ˇS ě 0.

We complete the argument with the simple observation that the j-th column vector of Sˇ can be written as the sum of |Φ_j| vectors whose elements are nonnegative and sum up to 1. The latter statement is valid for each column of ˇS and implies the existence of a doubly stochastic matrix S that satisfies ˇSλ_‹ “SλpBq. This, in turn, validates the identityλpYq “SλpBq for some S PDⁿ.

For an arbitrary set of orthonormal basis vectors tw₁, . . . , w_nuspanning Rⁿ, define the orthogonal matrix W :“ rw₁, . . . , w_ns. By Theorem 2.6 and Lemma 6.2, we then derive the majorization relation

diagpW^TYWqăλpW^TYWq “λpYqăλpBq.

Thus, the observation that the eigenvalues of any feasible matrix variable Y to problem (5.11) are majorized by the eigenvalues of B implies the compliance of Y with the

inequalities

where diag^Ó_jp¨q denotes the j-th largest diagonal element of the corresponding matrix.

Since this relation holds valid for arbitrary choices of orthonormal bases spanningRⁿ, this naturally includes the set of eigenvectors of A. In this respect, the integration of EVB based constraints such as (6.12) into ESC is redundant. By the arguments for Theorem 3.2 and Corollary 5.4, we further derive the same conclusion for ES and VL.

Even for the SDP relaxation with the smallest dimension, QAP_ms, it is sufficient to incorporate only a subset of the inequalities in (6.12). The distribution of the positive and negative eigenvalues of B provides the opportunity to construct a stronger and

more efficient version of (6.12). For relaxation frameworks that utilize the PSD splitting defined in (3.9), we show the following result.

Lemma 6.3. For the parameter matrixB of a given QAP instancepA, B, Cq, letpB_{`</sub>, B<sub>´</sub>q denote the PSD splitting defined in (3.9). Additionally, let r<sub>`} and r_´ denote the ranks of the matrices B_` and B_´, respectively. If incorporated into the corresponding instance of relaxation (4.32), then

imply the validity of all inequalities in (6.12).

Proof. Regarding the first r´´1 inequalities, 1ďl ăr´, the positive semidefiniteness of Y` “Y `Y´ and (6.14a) require

Furthermore, the orthogonality of the eigenvectors tp_iuimplies

@lP t1, . . . , nu: Finally, adding (6.15) and (6.14b) yields

l

By using (6.14a) and (6.14b), we realize a tighter version of the discussed bounding technique necessitating only rankpBq ´2 inequality constraints instead of the original n´1 conditions. At a first glance, the reduction of the framework by not more than n´rankpBq `1 linear inequality constraints may be hardly worth the effort of elaborating the specific implementation details. Nevertheless, the influence on the solving procedure should not be underestimated. Each of these inequalities introducesn<sup>2</sup>or`_n`1

2

˘coefficients to the actual SDP data, respectively. In regard to the memory management of the applied solver, the number of coefficients can be quite important for the performance of the solving procedure.

For the actual implementation of the discussed EVB cuts, there are more details that deserve our attention. As already described for GLB based constraints, alsoEVB based ones like (6.12) can be modified for different reformulations of the actual quadratic assignment problem. Reduction rules to derive appropriate reformulations have been elaborated, for example, in [34, 44, 88]. In the final version of their matrix lifting based SDP relaxation [27, MSDR₃], Ding and Wolkowicz applied their EVB based constraints to a projected reformulation of the QAP. By [27, Lemma 2.2], it was moreover shown that the corresponding relaxation incorporates the projection bound (PB) introduced in [44].

Hadley, Rendl, and Wolkowicz demonstrated in [44] thatPBoutperformsEVB1 for all tested QAP instances. In consideration of the interaction between the actual eigenvalue bound and the respective SDP relaxation in which this bound shall be incorporated, numerical tests for a wider range of problems taken from the QAP library [18] showed a slightly different picture. As a suitable integration in the respective SDP frameworks the author suggests the straightforward utilization of the reformulated QAP instance defined in (4.20). Actually, maybe not completely straightforward. The effect of the inequality conditions in (6.12) can be improved by a slight modification to our initial presuppositions on the eigenvalues and eigenvectors ofA and B. For this purpose, we exploit our knowledge about the presence of the particular eigenvector ^?1_ne. Since this vector is unaffected by permutations and the corresponding eigenvalue is equal to zero, it is possible to remove it from theEVB based inequalities. Let the index to this specific eigenvalue-eigenvector pair be fixed to i “ 1, and let all other eigenvalue-eigenvector pairs satisfy the general presuppositions for this Subsection. In this context,A and B

may be written as A“

ÿn i“2

µ_ip_ip^T_i , µ₂ ěµ₃ ě. . .ěµ_n, xe, p_iy “0|_2ďiďn (6.16a) and

B “

n

ÿ

i“2

λ_iq_iq_i^T, λ₂ ďλ₃ ď. . .ďλ_n, xe, q_iy “0|_2ďiďn, (6.16b) where µ1 “ λ1 “0 and p1 “q1 “ ^?1_ne. If we apply these adjusted index assignments, then

0ď

l

ÿ

i“2

p^T_i Yp_i´λ_i for l P t2, . . . , n´1u, (6.17) states a tighter and more economic version of (6.12).

For constraints of the form 0ď

l

ÿ

i“1

w_i^TYw_i´λ_i for l P t1, . . . , n´1u,

it is evident that the choice of the basis vectors tw₁, . . . , w_nu has a significant influence on the bounding quality. Considering the objective function xA,Yy ` xC,Xy, the choice of the eigenvectors of A is reasonable since it incorporates the corresponding eigenvalue bound. Nevertheless, this choice may not necessarily be the best possible one. A very similar argument as the one we used to explain the choice of the reformulation vector d_b given in (4.20) is also applicable to a reformulation of the matrix A.

The reformulation vectors d_a and v_a defined in (4.20) are designed to minimize the Frobenius norm of the reformulated data matrix A. For a strong eigenvalue bound this approach is reasonable but can be improved. The last statement is evident from superior performance of the bounding techniques PB [44] and EVB2 [88] compared to EVB1 [88].

Instead of simply taking over one of these approaches, we exploit the idea of weighted positive and negative semidefinite parts of A. More specifically, we utilize a splitting approach for A which is weighted in regard to the eigenvalue distribution of B. The corresponding adaptation of problem (4.18) is given by

d_a,v_aPRⁿinf, A1,A2PS<sub>`</sub><sup>n</sup> ~α1A1`α2A2~f

s.t. A`diag<sup>*</sup>pd<sub>a</sub>q `v_ae^T `ev_a^T “A1´A2,

(6.18)

where the weighting coefficients α₁ and α₂ are defined in respect of the eigenvalues ofB:

By solving problem (6.18), we obtain new reformulation vectors d_a and v_a. Since the eigenspace of the corresponding reformulation ofAis often more advantageous to compute tight eigenvalue bounds, we utilize the eigenvalue decomposition of

A´“A`diag<sup>*</sup>pd<sub>a</sub>q `v_ae^T `ev_a^T “ the ordering of the eigenvaluest´µ_iusatisfies our presuppositions in (6.16).

If the respective SDP relaxation is used within a branch-&-bound algorithm, it is possible to attain more beneficial sets of basis vectorstw1, . . . , wnuin a significantly more efficient way. The approach is as follows: suppose that the respective SDP relaxation has already been computed for different subproblems of the considered QAP. From the pool of already solved SDP relaxations, choose the instance which is most similar to the problem that needs to be solved in the current bounding step. Instead of the eigenvectors of the (possibly reformulated) coefficient matrix A, utilize the eigenvalue decomposition of the matrix ˆY obtained from the solution vector to the chosen problem instance. Order the eigenvectors with respect to the accompanied eigenvalues of ˆY and apply the necessary adaptations for the applicability to the current relaxation instance.

The latter step may involve the transformation into another space.

By allowing higher efforts on the implementation as well as the computations, it is possible to strengthen theEVB based cuts. In that context, let us consider the convex quadratic programming frameworkSOCPB introduced in [110]. For the construction of this relaxation, Xia uses the identity

trpAXBX^Tq “ tr

He defines a matrix S :“ ps_ijq with s_ij “ xp_i, Xq_jy² for 1 ď i, j ď n, and describes a relaxation of the corresponding quadratic equalities via

sij ě xp_i,Xq_jy², 1ďi, j ďn, (6.21) together with the equality constraints that realize S P Eⁿ. The latter condition is an immediate consequence of the orthogonality of tp_iuand tq_ju, yielding

@X PΠⁿ, j P t1, . . . , nu:

n

ÿ

i“1

xp_i, Xq_jy² “ }rp₁, . . . , p_ns^TXq_j}² “ }q_j}² “1 and

@X PΠⁿ, iP t1, . . . , nu:

n

ÿ

j“1

xp_i, Xq_jy² “ }p^T_i Xrq₁, . . . , q_ns}² “ }p_i}² “1.

For the integration into the respective SDP relaxation, we introduce the same matrix variable S, add the corresponding equality constraints for S P Eⁿ together with the inequalities in (6.21), and exploit the identities

p^T_i XBX^Tp_i “p^T_i X

˜ _n ÿ

j“1

λ_jq_jq^T_j

¸

X^Tp_i “

n

ÿ

j“1

λ_jxp_i,Xq_jy², 1ďi, j ďn,

to link the variables Y and S via the following equality conditions p^T_i Yp_i “

ÿn j“1

λjsij, 1ďi, j ďn. (6.22)

From the proof of Lemma 6.2, it is clear that the incorporation of these conditions into ESC, ES, or VL is redundant, at least if we assume ε“0. Additional upper bound constraints on the variables tsiju can change this. In order to attain a further tightening of the framework SOCPB, Xia utilizes the following linear upper bounds

@i, j P t1, . . . , nu: pl_ij `u<sub>ij</sub>qp<sup>T</sup><sub>i</sub> Xq<sub>j</sub> ´l<sub>ij</sub>u<sub>ij</sub> ěs<sub>ij</sub>, (6.23) where lij :“ xpi, qjy´ and uij :“ xpi, qjy` define lower and upper bounds of the corre-sponding linear terms tp^T_i Xq_ju, respectively.

We derive similar upper bounds as in (6.23) by exploiting the following identities

Together with the limits of the respective sum terms δ_ij^l :“ min

we obtain new linear bounding constraints:

maxtu_ij,´l_iju |p_i|^TX|q_i| `maxtu_ijδ_ij^u, l_ijδ_ij^l u ěsij, 1ďi, j,ďn. (6.25) In this context, it is worth mentioning that the necessary computations for the values defined in (6.24) can be realized very efficiently via

δ_ij^l “ xp^Ó_i, q^Ò_jy ` x|p^Ó_i|,|q^Ò_j|y and δ_ij^u “ xp^Ó_i, q_j^Óy ´ x|p^Ó_i|,|q^Ó_j|y.

Moreover, by introducing intermediate variables for the terms tX|q_j|u (alternatively t|p_i|^TXu), it is possible to reduce the number of nonzero coefficients that are necessary for the implementation of (6.25) to about 2n³.

Although (6.25) does not imply the validity of (6.23) - meaning that (6.25) is not strictly tighter than (6.23) - the former performs in general significantly better. This statement is particularly true if the respective constraints are incorporated into one of the discussed SDP frameworks.

If the computational costs are of minor importance, it is possible to use the even stronger upper bounds:

xM¯_ij,Xy `maxt¯δ_ij^u,δ¯_ij^l u ěsij, 1ďi, j ďn, (6.26) where ¯M_ij :“maxpl_ijp_iq_j^T, u_ijp_iq_j^Tq are defined as the element-wise maxima of the corre-sponding rank-1 parameter matrices, and

¯δ_ij^l :“ max

XPΠⁿxl_ijp_iq_j^T ´M¯_ij, Xy, δ¯_ij^u :“ max

XPΠⁿxu_ijp_iq^T_j ´M¯_ij, Xy

define the corresponding adaptations to the offset corrections in (6.24). The respective coefficient matrices still have low ranks, providing similar opportunities for the reduction of the computational costs like the ones we indicated for the implementation of (6.25).

Nevertheless, due to the absence of reiterations in the corresponding computations, the author has not been able to reduce the computational complexity below Opn³lognq. In respect of the small influence on the tightness of the considered SDP relaxations and the significantly greater computational effort, the constraints in (6.25) seem preferable to the ones in (6.26).

If we are concerned with larger QAP instances, even the constraints in (6.23) and (6.25) seem rather impractical. Though it is possible to realize a deep integration into ES and VL, the additional effort does not pay off in the same way for other relaxations frameworks. By combining the approach in (6.17) with some of the bounds in (6.25), it is possible to obtain a very efficient integration of the eigenvalue bound. Let t¯sijudenote upper bounds for the respective quadratic terms txp_i,Xq_jy²u. For the eigenvector p₂ of A, it is easy to see that

p^T₂XBX^Tp₂ ěλ₂xp₂,Xq₂y²`λ<sub>3</sub>`

1´ xp₂,Xq₂y²˘

ěλ₂s¯₂₂`λ₃p1´s¯₂₂q.

With ¯s2:3 :“mint¯s22,s¯33u, we educe the inequality for the sum over the first two terms:

3

ÿ

i“2

p^T_i XBX^Tp_i ěλ₂s¯_2:3`λ<sub>3</sub>p1´¯s<sub>2:3</sub>q `λ₂p1´s¯_2:3q `λ<sub>3</sub>s¯<sub>2:3</sub> “λ<sub>2</sub>`λ₃.

This matches the second inequality in (6.17). The third condition may then again be improved:

4

ÿ

i“2

p^T_i XBX^Tp_i ěλ₂`λ<sub>3</sub>`λ₄s₄₄`λ₅p1´s₄₄q.

It is therefore recommendable to replace every second inequality in (6.17) by pλ_l`1´λ_lqp1´¯s_llq ď

l

ÿ

i“2

p^T_i Yp_i´λ_i for l “2,4,6, . . . . (6.27) Appropriate terms for ¯s_ll can be taken from (6.23), (6.25), or even (6.26). For a minimal computational costs, one may simply use ¯s_ll“maxtu²_ll, l_ll²u.

Im Dokument Semidefinite Relaxation Approaches for the Quadratic Assignment Problem (Seite 97-107)