Abstract
The standard technique of reduced cost fixing from linear programming is not trivially extensible to semidefinite relaxations as the corresponding Lagrange multipliers are usually not available. We pro pose a general technique for computing reasonable Lagrange multipliers to constraints which are not part of the problem description. Its specialization to the semidefinite {-1,1} relaxation of quadratic 0-1 programming yields an efficient routine for fixing variables. The routine offers the possibility to exploit problem structure. We extend the traditional bijective map between {0,1} and {-1,1} formulations to the constraints such that the dual variables remain the same and structural properties are preserved. In consequence the fixing routine can efficiently be applied to optimal solutions of the semidefinite {0,1} relaxation of constrained quadratic 0-1 programming, as well. We provide numerical results showing the efficacy of the approach.
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F. Alizadeh, J.-P. A. Haeberly, and M. L. Overton. Coplementarity and nondegeneracy in semidefinite programming. Mathematical Programming, 77(2):111–128, 1997.
F. Barahona, M. Jünger, and G. Reinelt. Experiments in quadratic 0-1 programming. Mathematical Programming, 44:127–137, 1989.
C. De Simone. The cut polytope and the boolean quadric polytope. Discrete Applied Mathematics, 79:71–75, 1989.
M. X. Goemans and D. P. Williamson. Improved approxiamtion algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42:1115–1145, 1995.
G. H. Golub and C. F. van Loan. Matrix Computations. The Johns Hopkins University Press, 2nd edition, 1989.
M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, 2nd edition, 1988.
C. Helmberg, S. Poljak, F. Rendl, and H. Wolkowicz. Combining semidefinite and polyhedral relaxations for integer programs. In E. Balas and J. Clausen, editors, Integer Programming and Combinatorial Optimization, volume 920 of Lecture Notes in Computer Science, pages 124–134. Springer, May 1995.
C. Helmberg and F. Rendl. Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. ZIB Preprint SC-95-35, Konrad Zuse Zentrum für Informationstechnik Berlin, Takustraße 7, D-14195 Dahlem, Germany, Nov. 1995.
S. E. Karisch and F. Rendl. Semidefinite programming and graph equipartition. Technical Report 302, Department of Mathematics, Graz University of Technology, Graz, Austria, Dec. 1995.
M. Laurent, S. Poljak, and F. Rendl. Connections between semidefinite relaxations of the max-cut and stable set problems. Mathematical Programming, 77(2):225–246, 1997.
L. Lovász. On the Shannon capacity of a graph. IEEE Transactions on Information Theory, IT-25(1):1–7, Jan. 1979.
L. Lovász and A. Schrijver. Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optimization, 1(2):166–190, May 1991.
Y. Nesterov and M. J. Todd. Self-scaled barriers and interior-point methods for convex programming. Technical Report TR 1091, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853, Apr. 1994. Revised June 1995, to appear in Mathematics of Operations Research.
L. Tunçel. Primal-dual symmetry and scale invariance of interior-point algorithms for convex optimization. CORR Report 96-18, Department of Combinatorics and Optimization, Univeristy of Waterloo, Ontario, Canada, Nov. 1996.
A.C. Williams. Quadratic 0-1 programming using the roof dual with computational results. RUTCOR Research Report 8-85, Rutgers Unversity, 1985.
H. Wolkowicz and Q. Zhao. Semidefinite programming relaxations for the graph partitioning problem. Corr report, University of Waterloo, Ontario, Canada, Oct. 1996.
Q. Zhao, S. E. Karisch, F. Rendl, and H. Wolkowicz. Semidefinite programming relaxations for the quadratic assignment problem. CORR Report 95/27, University of Waterloo, Ontario, Canada, Sept. 1996.
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© 1997 Springer-Verlag Berlin Heidelberg
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Heimberg, C. (1997). Fixing variables in semidefinite relaxations. In: Burkard, R., Woeginger, G. (eds) Algorithms — ESA '97. ESA 1997. Lecture Notes in Computer Science, vol 1284. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63397-9_20
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DOI: https://doi.org/10.1007/3-540-63397-9_20
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