### Table 7.3: Nine previously unsolved benchmark instances taken from QAPLIB. The tree-size estimator enabled us to select those instances that could be solved in the available amount of time. After we had published our results, Tschoke et al. implemented a sim- ilar algorithm on their parallel PowerPC computer and solved the nug24 instance. Their result motivated us to develop a dynamic programming al- gorithm based on a new relaxation of the QAP which is stronger than the Gilmore{Lawler bound. By running this algorithm on the Paragon, we could

### Table 3.3 illustrates the strength of the SDP relaxation on several Nugent problems from QAPLIB compared to other bounds in the literature: Gilmore-Lawler bound (GLB) [6, 14], the projection or elimination bound ELI of [8], and the improved eigenvalue bound EVB3 from [26]. We note the high cost of the SDP bound for n=30 in the table and the low relative error for the bounds. A relaxed form of the SDP bound played a major role in the solution to optimality of several hard QAPs, see [3].

2002

### Table 1 is a comparison of bounds obtained from MSDR3 and other relaxation methods applied to instances from QAPLIB [6]. The first column OPT denotes the exact optimal value of the problem instance, while the following columns contain the lower bounds from the relaxation methods: GLB , the Gilmore-Lawler bound [10]; KCCEB , the dual linear programming bound [15]; P B , the projected eigenvalue bound [12]; QP B , the convex quadratic programming bound [1]; SDR1 , SDR2 , SDR3 , the vector-lifting semidefi- nite relaxation bounds [27] computed by the bundle method [24]; the last column is our MSDR3 . All output values are rounded up to the nearest integer. To solve QAP , the minimization of trace AXBXT and trace BXAXT are equivalent. But for the relaxation MSDR3 , exchanging the roles of A and B results in two different formulations and bounds. In our tests we use both versions and take the larger output as the bound of MSDR3 . We then keep the better formulation throughout the branch and bound process, so that we do not double the computational work.

2006

"... In PAGE 16: ... We then keep the better formulation throughout the branch and bound process, so that we do not double the computational work. From Table1 , we see that the relative performances between the LP -based bounds GLB , KCCEB are unpredictable. At some instances, both are weaker than even the least expensive P B bounds.... In PAGE 17: ...a. 4887 4965 4621 Nug30 6124 4539 4785 5266 5362 5413 5651 5803 5446 rou12 235528 202272 223543 200024 205461 208685 219018 223680 207445 rou15 354210 298548 323589 296705 303487 306833 320567 333287 303456 rou20 725522 599948 641425 597045 607362 615549 641577 663833 609102 scr12 31410 27858 29538 4727 8223 11117 23844 29321 18803 scr15 51140 44737 48547 10355 12401 17046 41881 48836 39399 scr20 110030 86766 94489 16113 23480 28535 82106 94998 50548 tai12a 224416 195918 220804 193124 199378 203595 215241 222784 202134 tai15a 388214 327501 351938 325019 330205 333437 349179 364761 331956 tai17a 491812 412722 441501 408910 415576 419619 440333 451317 418356 tai20a 703482 580674 616644 575831 584938 591994 617630 637300 587266 tai25a 1167256 962417 1005978 956657 981870 974004 908248 1041337 970788 tai30a 1818146 1504688 1565313 1500407 1517829 1529135 1573580 1652186 1521368 tho30 149936 90578 99855 119254 124286 125972 134368 136059 122778 Table1 : Comparison of bounds for QAPLIB instances... ..."

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### Table 5: Results for rst level in branching tree

"... In PAGE 33: ...rst level of the branching tree, i.e. we want to see how fast the lower bounds increase after branching. Table5 gives the results for the rst level, when one partial assignment is made. As pointed out above, the Nugxx examples posses inherent symmetries due to their distance matrices, e.... In PAGE 33: ... To measure the increasing rate (ir) of the lower bound (lbd) by branching, we de ne the rate in percent as follows. ir := lbdchild ? lbdparent lbdparent 100% In Table5 , the increasing rates are shown by the numbers in the brackets. The results of this table show that the lower bounds given by the SDP relaxations increase much faster than the Gilmore-Lawler bounds in the rst level of the branching tree.... ..."

### Table 1: Comparison of bounds for QAPLIB instances

2006

"... In PAGE 16: ... 3 Numerical Results 3.1 Comparing Bounds for QAPLIB Problems Table1 is a comparison of bounds obtained from MSDR3 and other relaxation methods applied to instances from QAPLIB [6]. The first column OPT denotes the exact optimal value of the problem instance, while the following columns contain the lower bounds from the relaxation methods: GLB , the Gilmore-Lawler bound [10]; KCCEB , the dual linear programming bound [15]; P B , the projected eigenvalue bound [12]; QP B , the convex quadratic programming bound [1]; SDR1 , SDR2 , SDR3 , the vector-lifting semidefi- nite relaxation bounds [27] computed by the bundle method [24]; the last column is our MSDR3 .... In PAGE 16: ... We then keep the better formulation throughout the branch and bound process, so that we do not double the computational work. From Table1 , we see that the relative performances between the LP -based bounds GLB , KCCEB are unpredictable. At some instances, both are weaker than even the least expensive P B bounds.... ..."

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### TABLE 15. CVT Reaction Bottleneck of R1 in SES and ESP Approximations at 298 K

1999

Cited by 1

### Table 1: The classical axioms

1996

"... In PAGE 4: ... a-free) expressions. Examples of equations in the theory S are those in Table1 , called the classical axioms by Conway [6, page 25], and the laws xy = yx (x + y) = x y : Unlike the classical axioms, the laws above only hold under the assumption that the alphabet is a singleton. The following identity is an easy consequence of the classical axioms: 0 = 1 : (1) An example of an equation that is contained in E, but not in S, is a + x = a :... In PAGE 6: ...Table1 . Instantiating these equations, we derive that EF proves all of the equalities C14:n(a).... In PAGE 8: ... In that case, we shall simply write Mp[[P ]] for the denotation of P in the algebra Mp. It is not hard to see that the equations C1{13 in Table1 are sound in the algebra Mp. We now proceed to show that the algebra Mp meets the requirements P1 and P2 that we set out to achieve.... ..."

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### Table 1: Classical approaches

"... In PAGE 3: ... Of course, the NoP can become a crucial factor, since a large number results in high test costs. In Table1 the name of the benchmark is given in the first column followed by the number of inputs and outputs in column two and three, respectively. The number of literals, the number of paths and the PDFC are given in column lits, NoP and PDFC, respectively.... ..."

### Table 1: Classical algorithm

### Table 1: Classical radial basis functions 16 Classical RBF Equation

2003

"... In PAGE 4: ... (35) Some of the classical radial functions used in multivariate interpolation are presented in Table 1. Note that the shape parameter c in the radial basis functions in Table1 is user-defined and can be adjusted Table 1: Classical radial basis functions 16 Classical RBF Equation ... ..."

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