Abstract
Copositive optimization problems are particular conic programs: optimize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone from within, we propose a method which approximates this cone from outside. This is achieved by passing to the dual problem, where the feasible set is an affine subspace intersected with the cone of completely positive matrices, and this cone is approximated from within. We consider feasible descent directions in the completely positive cone, and regularized strictly convex subproblems. In essence, we replace the intractable completely positive cone with a nonnegative cone, at the cost of a series of nonconvex quadratic subproblems. Proper adjustment of the regularization parameter results in short steps for the nonconvex quadratic programs. This suggests to approximate their solution by standard linearization techniques. Preliminary numerical results on three different classes of test problems are quite promising.
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References
Berman A., Shaked-Monderer N.: Completely positive matrices. World Scientific, Singapore (2003)
Bomze I.M., Dür M., de Klerk E., Quist A., Roos C., Terlaky T.: On copositive programming and standard quadratic optimization problems. J. Glob. Optim. 18, 301–320 (2000)
Bomze, I.M., Eichfelder, G.: Copositivity detection by difference-of-convex decomposition and omega-subdivision. Available at http://www.optimization-online.org/DB_HTML/2010/01/2523.html (2010)
Bomze I.M., Frommlet F., Locatelli M.: Gap, cosum, and product properties of the θ′ bound on the clique number. Optimization 59, 1041–1051 (2010)
Bomze I.M., Jarre F.: A note on Burer’s copositive representation of mixed-binary QPs. Optim. Lett. 4, 465–472 (2010)
Bundfuss S., Dür M.: Algorithmic Copositivity Detection by Simplicial Partition. Linear Algebra Appl. 428, 1511–1523 (2008)
Bundfuss S., Dür M.: An Adaptive Linear Approximation Algorithm for Copositive Programs. SIAM J. Optim. 20, 30–53 (2009)
Burer S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120, 479–495 (2009)
de Klerk E., Pasechnik D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12, 875–892 (2002)
Dickinson P.J.C.: An improved characterization of the interior of the completely positive cone. Electron. J. Linear Algebra 20, 723–729 (2010)
DIMACS implementation challenges. Available at ftp://dimacs.rutgers.edu/pub/challenge/graph
Dür M., Still G.: Interior points of the completely positive cone. Electron. J. Linear Algebra 17, 48–53 (2008)
Dukanovic I., Rendl F.: Copositive programming motivated bounds on the stability and chromatic numbers. Math. Program. 121, 249–268 (2010)
Johnson C.R., Reams R.: Constructing copositive matrices from interior matrices. Electron. J. Linear Algebra 17, 9–20 (2008)
Knuth D.E.: The sandwich theorem. Electron. J. Comb. 22, 1–48 (1994)
Lovász L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory IT–25, 1–7 (1979)
McEliece R.J., Rodemich E.R., Rumsey H.C.: The Lovász’ bound and some generalizations. J. Comb. Inf. Syst. Sci. 3, 134–152 (1978)
Nesterov Y., Nemirovski A.: Interior point polynomial algorithms in convex programming. SIAM Publications, Philadelphia (1994)
Parrilo, P.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, USA (2000)
Povh J., Rendl F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discret. Optim. 6, 231–241 (2009)
Schrijver A.: A comparison of the Delsarte and Lovasz bounds. IEEE Trans. Inf. Theory IT–25, 425–429 (1979)
Vandenbussche D., Nemhauser G.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102, 559–575 (2005)
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Bomze, I.M., Jarre, F. & Rendl, F. Quadratic factorization heuristics for copositive programming. Math. Prog. Comp. 3, 37–57 (2011). https://doi.org/10.1007/s12532-011-0022-z
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DOI: https://doi.org/10.1007/s12532-011-0022-z