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.
KeywordsConvex Cone Quadratic Constraint Dual Cone Complementarity Constraint Positive Matrice
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.
- 1.Anstreicher, K.M., Burer, S.: Computable representations for convex hulls of low-dimensional quadratic forms. Mathematical Programming (series B) 124, 33–43 (2010)Google Scholar
- 4.Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific (2003)Google Scholar
- 6.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)Google Scholar
- 12.Burer, S., Dong, H.: Separation and relaxation for cones of quadratic forms. Manuscript, University of Iowa (2010) Submitted to Mathematical Programming.Google Scholar
- 14.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)Google Scholar
- 15.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.
- 16.Dukanovic, I., Rendl, F.: Copositive programming motivated bounds on the stability and the chromatic numbers. Math. Program. 121(2, Ser. A), 249–268 (2010)Google Scholar
- 17.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)Google Scholar
- 18.Dür, M., Still, G.: Interior points of the completely positive cone. Electron. J. Linear Algebra 17, 48–53 (2008)Google Scholar
- 19.Eichfelder, G., Jahn, J.: Set-semidefinite optimization. Journal of Convex Analysis 15, 767–801 (2008)Google Scholar
- 20.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)Google Scholar
- 21.Faye, A., Roupin, F.: Partial lagrangian relaxation for general quadratic programming. 4OR 5, 75–88 (2007)Google Scholar
- 22.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)Google Scholar
- 26.Maxfield, J.E., Minc, H.: On the matrix equation X ′ X = A. Proc. Edinburgh Math. Soc. (2) 13, 125–129 (1962/1963)Google Scholar
- 30.Parrilo, P.: Structured Semidefinite Programs and Semi-algebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology (2000)Google Scholar
- 31.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)Google Scholar
- 32.Povh, J.: Towards the optimum by semidefinite and copositive programming : new approach to approximate hard optimization problems. VDM Verlag (2009)Google Scholar
- 36.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)Google Scholar
- 37.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.Google Scholar