Matrix lifting revisited

In the last subsection, we described reduction schemes that serve the reduction of the dimension of SDP constraints. For this purpose, we exploited our knowledge about the eigenspace of B and tried to obtain more beneficial sets of eigenvaluestλ˜₁, . . . ,˜λ_nu via approximations of the original parameter matrices. In consideration of the matrix lifting strategy, it is possible to exploit a low rank of B in quite the opposite way. In the following, we will describe a possibility to utilize the presence of a non-trivial nullspace ofB, not for a reduction but for a tightening of the respective semidefiniteness condition.

We follow the index set definitions from the last subsection and utilize the compact eigenvalue decompositionB “Q_Ω0Λ_Ω0Q^T_Ω0 to establish the following identity:

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The semidefiniteness property of the left-hand side serves as a basis for a new relaxation framework. In this context, we first investigate the usability of the conditions

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and as a replacement for the SDP constraint of ML₁.

If the rank of B is a good deal smaller than n, this substitution has only a slight effect on the overall computational effort. In rare cases, the reduced dimension of the matrix in (5.13a) can actually speed up the solving procedure. The more important matter is how this modification affects the quality of the matrix lifting based relaxation framework. The matrix variable G in (5.13a) and (5.13b) is used to relax the quadratic term XQ_Ω0Q^T_Ω0X^T. With the following result, we show that the presented substitution tightens the relaxation.

Lemma 5.1. Any quadruple of matrices pG,X,Y,Zq that satisfies the semidefiniteness conditions in (5.13a) and (5.13b) also complies with constraint (3.5b).

Proof. Define the two block-diagonal matrices D_yz :“ diag^*pQ_Ω⁰, I_p2nqq and D_g :“

is identical to the matrix in (3.5b). The positive definiteness of this matrix is a direct consequence of (5.13a) and (5.13b).

Apparently, a smaller rank of matrix B decreases the trace of variable G. An decreased trace leads to a stronger SDP constraint which thereby improves the quality of the relaxation. For QAP instances with low-rank parameter matricesB, the improvement can be immense and is absolutely worth the slightly increased computational effort accompanied by the replacement of (4.31b) with (5.13).

One way to exploit this correlation beyond the already mentioned modifications is the utilization of a low-rank approximation ˜B “Q_ΩεΛ_ΩεQ_Ωε. Similarly to the approach used forQAP_esc, however, the corresponding residual R“B´B˜ requires a special treatment.

This includes the possible drawbacks accompanied by the selected approach. Although

the tighter bound for the significant term trpAXBX˜ ^Tq usually outweighs the possibly weaker relaxation of the remainder term, the improvements turn out to be relatively small, whereas the computational effort increases significantly. For now, we therefore dismiss the idea of constraint splittings based on low-rank approximation.

It is important to realize that the conclusion from above does by no means apply to the general idea of matrix splitting based rank reductions. By utilization of the Schur complement inequality to the respective matrix blocks, we see that (5.13a) involves the relation Z ľYG^:Y. This inequality indicates a strong (nearly proportional) correlation between the expression~G¹⁴Z¹²G¹⁴ ´XBX^T~ and the tightness of the semidefiniteness condition in (5.13a). By a loose interpretation of this connection, one may conclude that smaller values of trpGqtrpZq “ ~BB^:~²_f~B~²_f indicate stronger relaxation bounds. In this regard, we are looking for a new splitting scheme B “ B₁ `B₂ with the aim of minimizing the sum of the corresponding product termsř2

i“1~B_iB_i^:~²_f~B_i~²_f. In order to achieve this, we apply a reverse optimization of the respective factors. This means that we concentrate on the minimization of the factors ~B₁B₁^:~²_f and ~B₂~²_f. In the actual implementation, we use a splitting scheme based on the spectral value decomposition of B. The splitting is realized in such a way that B₁ contains the most significant eigenvalues ofB, but relatively few compared to the overall number of eigenvalues. The remainder part B₂, on the other hand, contains more eigenvalues with smaller absolute values. The individual application of the semidefiniteness condition in (5.13a) to each of these matrices results in a significant strengthening of the relaxation.

Other ideas for efficiency and quality improvements over the original relaxation framework QAP_ml are based on the reformulation and reduction strategies discussed in Section 4.1 as well as the approximation approach used in (5.6). We combine the addressed modifications and construct an extended SDP framework based on the matrix

lifting approach denoted QAP_mlx or simply MLX: intersection of Ω^ε and the respective index sets which describe the eigenvalue assignment to the parts B₁ and B₂.

If the considered QAP instance requires a reduction of the computational expense, one may decrease the approximation threshold ε and replace the semidefinite constraints in (5.14c) by I´G1´G2 P S_`. The complete removal of this constraint is generally not advisable. For many problem instances, the splitting scheme from above provides the opportunity for a further framework reduction. The described minimization procedure leads to a small rank of B₁. Often this matrix contains only a single nonzero eigenvalue or a single cluster of nonzero eigenvalues. If this is the case, we may replace D₁ with D˜₁ :“diag^*pQ_Ω^ε

1, I_p2n,nqqand setY1 “ _rankpB^trpB1q

1qG1. Even if there are two clusters of nonzero eigenvalues in B₁, it can be beneficial to replace the first constraint in (5.14b) with the respective semidefiniteness conditions based on the ESC approach.