Q. ZHAO. Semidefinite Programming for Assignment and Partitioning Problems. PhD thesis, University of Waterloo, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....programming approaches. Whether this approach can be extended to other problems is an interesting question. Some work on attempting to exploit sparsity in the general setting has been performed by Fujisawa et al. 38] and by Helmberg et al. 56] in their MATLAB implementation. Zhao et al. [152, 151] propose using a preconditioned conjugate gradient method to calculate the directions for the quadratic assignment problem (QAP) within a primal infeasible dual feasible variant of the method proposed in [56] In the setting of solving a QAP, the semidefinite relaxation is used to obtain a lower ....

Q. Zhao. Semidefinite programming for assignment and partitioning problems. PhD thesis, Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada, 1996.

....this is a computationally difficult optimization problem, so it is important to have tight tractable relaxations. A direct application of the generic imbedding would lead to a semidefinite problem with matrices of order nk Theta nk: This approach is investigated in some detail in the thesis [64] by Zhao. However, the partition matrices Y are highly structured (0; 1) Gammamatrices, so exploiting this structure should provide some additional information to get good relaxations. Here we mention the following properties of partition matrices Y : 4 1. Y t Y = mI k (columns pairwise ....

....scientific discussion going on, as to how this system should be solved to maintain numerical stability. The following choices are currently proposed. ffl (partial) Schur complements and backsubstitution; see for instance [2, 29] ffl Least Squares approach; see [36] ffl Iterative methods; see [64]. At present, there is no clear winner as to what is the most efficient way to solve the system. The conflicting goals are computational efficiency versus numerical accuracy and stability. In [59] computational evidence is provided that the NT direction achieves higher accuracy than both the ....

Q. ZHAO. Semidefinite Programming for Assignment and Partition Problems. PhD thesis, University of Waterloo, 1996. 27

....above mentioned Euclidean space of matrices. Successful applications of semidefinite programming in discrete optimization are presented in Goemans and Williamson [82] and Lov asz and Schrijver [125] Recently, semidefinite programming relaxations for the QAP were considered by Karisch [103] Zhao [176], and Zhao, Karisch, Rendl and Wolkowicz [177] The SDP relaxations considered in these papers are solved by interior point methods or cutting plane methods, and the obtained solutions are valid lower bounds for the QAP. In terms of quality the bounds obtained in this way are competitive with the ....

Q. Zhao, Semidefinite Programming for Assignment and Partitioning Problems, Ph.D. Thesis, University of Waterloo, Ontario, Canada, 1996.

.... these interior algorithms, see Anstreicher and Fampa [4] Alizadeh, Haeberly, and Overton [3] Fujisawa, Kojima and Nakata [8] Helmberg, Rendl, Vanderbei, and Wolkowicz [13] Karisch, Rendl, and Clausen [14] Vandenberghe and Boyd [31] Wolkowicz and Zhao [32] Zhao, Karisch, Rendl, and Wolkowicz [35][36] To the best of our knowledge, the largest problem that could be solved was at n = 900 from their reports. After the initial version of this paper was submitted, one more implementation came out: Fujisawa, Fukuda, Kojima and Nakata [10] reported that they could solve a maximum cut ....

....and dual objective values and the relative duality gap at termination. Also shown is the dimension and the percentage of nonzero entries in the objective matrix, as well as the time (in seconds) and number of iterations required by the program. Most of the previous numerical tests [7] 17] 32][35][36] were conducted on smaller problem data sets where the dimension n was only a few hundred or less, so that no available computation result could be compared to ours. After our results reported, a study of using a primal dual algorithm for solving relative larger problems, including the ....

Q. Zhao, "Semidefinite Programming for Assignment and Partitioning Problems" PhD thesis, University of Waterloo, 1996.

....programming approaches. Whether this approach can be extended to other problems is an interesting question. Some work on attempting to exploit sparsity in the general setting has been performed by Fujisawa et al. 38] and by Helmberg et al. 56] in their MATLAB implementation. Zhao et al. [152, 151] propose using a preconditioned conjugate gradient method to calculate the directions for the quadratic assignment problem (QAP) within a primal infeasible dual feasible variant of the method proposed in [56] In the setting of solving a QAP, the 62 J. E. MITCHELL, P. M. PARDALOS, AND M. G. C. ....

Q. Zhao. Semidefinite programming for assignment and partitioning problems. PhD thesis, Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada, 1996.

....papers in the bibliography [1, 2, 5, 12, 13, 18] The latex bib file can be obtained over WWW or with anonymous ftp using URL: ftp: orion.uwaterloo.ca pub henry reports setpart.bib.gz. In this paper, we develop an SDP relaxation for the set partitioning problem. Our approach is similar to that in [21, 22, 20]; i.e. we derive a semidefinite relaxation from the dual of the dual of a quadratic constrained quadratic program formulation of (SP) we employ a gangster operator to efficiently model the 0 1 constraints in the relaxation; we project the feasible set onto the minimal face of the semidefinite ....

.... 234 17327 18324 small04 33 192 584 4503 4503 small05 44 277 770 21706 21706 tiny04 6 27 72 1035 1091 tiny01 3 6 9 17.5 25.00 tiny05 7 35 70 1215 1257 Table 1: Numerical Results 3 NUMERICAL TESTS The algorithm (a p d i p approach) we use to solve the SDP relaxation is very similar to the one in [21, 22, 20] for the quadratic assignment and graph partitioning problems. An incomplete conjugate gradient method is used to solve the large Newton equations that arise at each iteration of the algorithm. As we have seen from the geometrical discussion above, the algorithm may have to deal with those ....

Q. ZHAO. Semidefinite Programming for Assignment and Partitioning Problems. PhD thesis, University of Waterloo, 1996. URL: ftp://orion.uwaterloo.ca/pub/henry/software/qap.d/zhaophdthesis.ps.gz.

.... quadratic problem is w (E cut ) min 1 2 trace X t LX subject to X ffi X = X kXe k Gamma e n k 2 = 0 kX t e n Gamma mk 2 = 0 X :i ffi X :j = 0 8i 6= j: To derive the semidefinite relaxation, we can now take the dual of the (homogenized) Lagrangian dual of this problem (see [13, 12] for the details of this approach applied to the quadratic assignment problem) A direct approach uses the substitution YX : 1 vec (X) 1 vec (X) t ) where vec (X) is the vector formed from the columns of X and YX 0; i.e. is positive semidefinite. For example, the objective function ....

....) Since both I k Gamma1 E k Gamma1 and I n Gamma1 E n Gamma1 are positive definite, we can see that when Gammaff is large enough V t W V is negative definite. 2 5 NUMERICAL TESTS The algorithm (a p d i p approach) we use to solve the SDP relaxation is very similar to the one in [13, 12] for the quadratic assignment problem. An incomplete conjugate gradient method is used to solve the large Newton equations that arise. After solving the relaxation, we obtain not only a lower bound for the graph partitioning problem but also an appropriate solution Y for the SDP relaxation. By ....

Q. ZHAO. Semidefinite Programming for Assignment and Partitioning Problems. PhD thesis, University of Waterloo, 1996.

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Q. ZHAO. Semidefinite Programming for Assignment and Partitioning Problems. PhD thesis, University of Waterloo, 1996.

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