Abstract
This chapter provides an introduction to copositive programming, which is linear programming over the convex conic of copositive matrices. Researchers have shown that many NP-hard optimization problems can be represented as copositive programs, and this chapter recounts and extends these results.
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Notes
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If \(\mathcal{K}\) is a semidefinite cone, than \(\mathcal{K}\) must be encoded in its “vectorized” form to match the development in this chapter. For example, the columns of a d ×d semidefinite matrix could be stacked into a single, long column vector of size d2, and then the CP matrices yyT over \({\mathfrak{R}}_{+} \times \mathcal{K}\) would have size \(({d}^{2} + 1) \times ({d}^{2} + 1)\).
References
Anstreicher, K.M., Burer, S.: Computable representations for convex hulls of low-dimensional quadratic forms. Mathematical Programming (series B) 124, 33–43 (2010)
Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization: Analysis, algorithms, and engineering applications. MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2001)
Berman, A., Rothblum, U.G.: A note on the computation of the CP-rank. Linear Algebra Appl. 419, 1–7 (2006)
Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific (2003)
Bomze, I., Jarre, F.: A note on Burer’s copositive representation of mixed-binary QPs. Optimization Letters 4, 465–472 (2010)
Bomze, I.M., de Klerk, E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. Dedicated to Professor Naum Z. Shor on his 65th birthday. J. Global Optim. 24(2), 163–185 (2002)
Bomze, I.M., Dür, M., de Klerk, E., Roos, C., Quist, A.J., Terlaky, T.: On copositive programming and standard quadratic optimization problems. J. Global Optim. 18(4), 301–320 (2000)
Bundfuss, S., Dür, M.: An adaptive linear approximation algorithm for copositive programs. SIAM J. Optim. 20(1), 30–53 (2009)
Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Mathematical Programming 120, 479–495 (2009)
Burer, S.: Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Mathematical Programming Computation 2(1), 1–19 (2010)
Burer, S., Anstreicher, K.M., Dür, M.: The difference between 5 ×5 doubly nonnegative and completely positive matrices. Linear Algebra Appl. 431(9), 1539–1552 (2009)
Burer, S., Dong, H.: Separation and relaxation for cones of quadratic forms. Manuscript, University of Iowa (2010) Submitted to Mathematical Programming.
de Klerk, E., Pasechnik, D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875–892 (2002)
de Klerk, E., Sotirov, R.: Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem. Math. Programming 122(2, Ser. A), 225–246 (2010)
Dong, H., Anstreicher, K.: Separating Doubly Nonnegative and Completely Positive Matrices. Manuscript, University of Iowa (2010) To appear in Mathematical Programming. Available at http://www.optimization-online.org/DB_HTML/2010/03/2562.html.
Dukanovic, I., Rendl, F.: Copositive programming motivated bounds on the stability and the chromatic numbers. Math. Program. 121(2, Ser. A), 249–268 (2010)
Dür, M.: Copositive Programming—A Survey. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds.) Recent Advances in Optimization and its Applications in Engineering, pp. 3–20. Springer (2010)
Dür, M., Still, G.: Interior points of the completely positive cone. Electron. J. Linear Algebra 17, 48–53 (2008)
Eichfelder, G., Jahn, J.: Set-semidefinite optimization. Journal of Convex Analysis 15, 767–801 (2008)
Eichfelder, G., Povh, J.: On reformulations of nonconvex quadratic programs over convex cones by set-semidefinite constraints. Manuscript, Faculty of Information Studies, Slovenia, December (2010)
Faye, A., Roupin, F.: Partial lagrangian relaxation for general quadratic programming. 4OR 5, 75–88 (2007)
Gvozdenović, N., Laurent, M.: Semidefinite bounds for the stability number of a graph via sums of squares of polynomials. In Lecture Notes in Computer Science, 3509, pp. 136–151. IPCO XI (2005)
Gvozdenović, N., Laurent, M.: Computing semidefinite programming lower bounds for the (fractional) chromatic number via block-diagonalization. SIAM J. Optim. 19(2), 592–615 (2008)
Gvozdenović, N., Laurent, M.: The operator Ψ for the chromatic number of a graph. SIAM J. Optim. 19(2), 572–591 (2008)
Hiriart-Urruty, J.-B., Seeger, A.: A variational approach to copositive matrices. SIAM Review, 52(4), 593–629 (2010)
Maxfield, J.E., Minc, H.: On the matrix equation X′X = A. Proc. Edinburgh Math. Soc. (2) 13, 125–129 (1962/1963)
Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Programming 39(2), 117–129 (1987)
Natarajan, K., Teo, C., Zheng, Z.: Mixed zero-one linear progams under objective uncertainty: A completely positive representation. Operations Research, 59(3), 713–728, (2011)
Nesterov, Y.E., Nemirovskii, A.S.: Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics, Philadelphia (1994)
Parrilo, P.: Structured Semidefinite Programs and Semi-algebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology (2000)
Peña, J., Vera, J., Zuluaga, L.F.: Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim. 18(1), 87–105 (2007)
Povh, J.: Towards the optimum by semidefinite and copositive programming : new approach to approximate hard optimization problems. VDM Verlag (2009)
Povh, J., Rendl, F.: A copositive programming approach to graph partitioning. SIAM J. Optim. 18(1), 223–241 (2007)
Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discrete Optim. 6(3), 231–241 (2009)
Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28(2), 246–267 (2003)
Wen, Z., Goldfarb, D., Yin, W.: Alternating direction augmented lagrangian methods for semidefinite programming. Manuscript, Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, USA (2009)
Yildirim, E.A.: On the accuracy of uniform polyhedral approximations of the copositive cone. Manuscript, Department of Industrial Engineering, Bilkent University, 06800 Bilkent, Ankara, Turkey (2009) To appear in Optimization Methods and Software.
Zhao, X., Sun, D., Toh, K.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20, 1737–1765 (2010)
Acknowledgements
The author wishes to thank Mirjam Dür and Janez Povh for stimulating discussions on the topic of this chapter. The author also acknowledges the support of National Science Foundation Grant CCF-0545514.
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Burer, S. (2012). Copositive Programming. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_8
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