Abstract
Finding global optimum of a non-convex quadratic function is in general a very difficult task even when the feasible set is a polyhedron. We show that when the feasible set of a quadratic problem consists of orthogonal matrices from \({\mathbb{R}^{n\times k}}\) , then we can transform it into a semidefinite program in matrices of order kn which has the same optimal value. This opens new possibilities to get good lower bounds for several problems from combinatorial optimization, like the Graph partitioning problem (GPP), the Quadratic assignment problem (QAP) etc. In particular we show how to improve significantly the well-known Donath-Hoffman eigenvalue lower bound for GPP by semidefinite programming. In the last part of the paper we show that the copositive strengthening of the semidefinite lower bounds for GPP and QAP yields the exact values.
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References
Anstreicher K.: Recent advances in the solution of quadratic assignment problems. Math. Program. B 97, 27–42 (2003)
Alpert C.J., Kahng A.B.: Recent directions in netlist partition: a survey. Integr. VLSI J. 19, 1–81 (1995)
Anstreicher K., Wolkowicz H.: On lagrangian relaxation of quadratic matrix constraints. SIAM J. Matrix Anal. Appl. 22, 41–55 (2000)
Arora S., Karger D., Karpinski M.: Polynomial time approximation schemes for dense instances of NP-hard problems. J. Comput. Syst. Sci. 58(1), 193–210 (1999)
Beck A.: Quadratic matrix programming. SIAM J. Optim. 17, 1224–1238 (2007)
Bomze I., Duer M., de Klerk E., Roos C., Quist A.J., Terlaky T.: On copositive programming and standard quadratic optimization problems. J. Glob. Optim. 18, 301–320 (2000)
Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program., published online on April 29th (2008)
Burer S., Vandenbussche D.: Solving lift-and-project relaxations of binary integer programs. SIAM J. Optim. 16, 726–750 (2006)
Burkard, R.E., Cela, E., Karisch, S.E., Rendl, F.: QAPLIB—a quadratic assignment problem library. Available at http://www.seas.upenn.edu/qaplib/, July (2007)
de Klerk E., Pasechnik D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12, 875–892 (2002)
de Klerk, E., Sotirov, R.: Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem. To appear in Math. Program., available on http://www.springerlink.com/content/85302n245v250051/fulltext.pdf.
Donath W.E., Hoffman A.J.: Lower bounds for the partitioning of graphs. IBM J. Res. Dev. 17, 420–425 (1973)
Falkner J., Rendl F., Wolkowicz H.: A computational study of graph partitioning. Math. Program. 66(2), 211–240 (1994)
Garey M.R., Johnson D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco (1979)
Guttmann-Beck N., Hassin R.: Approximation algorithms for minimum K-cut. Algorithmica 27, 198–207 (2000)
Helmberg C., Rendl F., Mohar B., Poljak S.: A spectral approach to bandwidth and separator problems in graphs. Linear Multilinear Algebra 39, 73–90 (1995)
Hoffman A.J., Wielandt H.W.: The variation of the spectrum of a normal matrix. Duke Math. J. 20, 37–39 (1953)
Karisch, S.E., Rendl, F.: Semidefinite programming and graph equipartition. In: Topics in Semidefinite and Interior-point Methods, The Fields Institute for Research in Mathematical Sciences, vol. 18. Communications Series, American Mathematical Society, Providence, RI (1998)
Lisser A., Rendl F.: Graph partitioning using linear and semidefinite programming. Math. Program. 95(1), 91–101 (2003)
Motzkin T.S., Straus E.G.: Maxima for graphs and a new proof of a theorem of Túran. Can. J. Math. 17, 533–540 (1965)
Papadimitriou C.H., Steiglitz K.: Combinatorial Optimization: Algorithms and Complexity. Dover Publications, Mineola, NY (1998)
Povh, J.: Application of semidefinite and copositive programming in combinatorial optimization. PhD Thesis, University of Ljubljana, Ljubljana (2006)
Povh J., Rendl F.: A copositive programming approach to graph partitioning. SIAM J. Optim. 18, 223–241 (2007)
Povh J., Rendl F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discrete Optim. 6, 231–241 (2009)
Rendl F., Sotirov R.: Bounds for the quadratic assignment problem using the bundle method. Math. Program. 109(2–3), 505–524 (2007)
Rendl F., Wolkowicz H.: A projection technique for partitioning the nodes of a graph. Ann. Oper. Res. 58, 155–179 (1995)
Wolkowicz H., Zhao Q.: Semidefinite programming relaxations for the graph partitioning problem. Discrete Appl. Math. 96–97, 461–479 (1999)
Zhao Q., Karisch S.E., Rendl F., Wolkowicz H.: Semidefinite programming relaxations for the quadratic assignment problem. J. Comb. Optim. 2, 71–109 (1998)
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Supported by the Slovenian Research Agency (project no. 1000-08-210518).
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Povh, J. Semidefinite approximations for quadratic programs over orthogonal matrices. J Glob Optim 48, 447–463 (2010). https://doi.org/10.1007/s10898-009-9499-7
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DOI: https://doi.org/10.1007/s10898-009-9499-7
Keywords
- Quadratic programming
- Semidefinite programming
- Copositive programming
- Eigenvalue bound
- Quadratic assignment problem
- Graph partitioning problem